Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
7-x-2*x^2
-
y=x^3
-
(3*x-4)^40/(x^2-2)^36
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y=x^2-x
-
Expresiones idénticas
- log10(x)-arctg(x)
- logaritmo de 10(x) menos arctg(x)
- log10x-arctgx
-
Expresiones semejantes
- log10(x)+arctg(x)
-
Expresiones con funciones
- arctg
- arctg(x)/(e^x)
- arctg(x)+5
- arctg(1\(x-1))
- arctg(-0.05x/20)
- arctg(x^2-1)
Gráfico de la función y = log10(x)-arctg(x)
Solución
Ha introducido
[src]
log(x)
f(x) = ------- - atan(x)
log(10)
$$f{\left(x \right)} = \frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}$$
f = log(x)/log(10) - atan(x)
Gráfico de la función
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 34.8420115290269$$
o sea hay que resolver la ecuación:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 34.8420115290269$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{1}{x^{2} + 1} + \frac{1}{x \log{\left(10 \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
$$x_{2} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Signos de extremos en los puntos:
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Decrece en los intervalos
$$\left(-\infty, - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right] \cup \left[\frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \infty\right)$$
Crece en los intervalos
$$\left[- \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{1}{x^{2} + 1} + \frac{1}{x \log{\left(10 \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
$$x_{2} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Signos de extremos en los puntos:
/ _______________\
| / 2 |
_______________ |log(10) \/ -4 + log (10) | / _______________ \
/ 2 log|------- - ------------------| | / 2 |
log(10) \/ -4 + log (10) \ 2 2 / |\/ -4 + log (10) log(10)|
(------- - ------------------, --------------------------------- + atan|------------------ - -------|)
2 2 log(10) \ 2 2 / / _______________ \
| / 2 |
_______________ / _______________ \ |\/ -4 + log (10) log(10)|
/ 2 | / 2 | log|------------------ + -------|
\/ -4 + log (10) log(10) |\/ -4 + log (10) log(10)| \ 2 2 /
(------------------ + -------, - atan|------------------ + -------| + ---------------------------------)
2 2 \ 2 2 / log(10) Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}$$
Decrece en los intervalos
$$\left(-\infty, - \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right] \cup \left[\frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \infty\right)$$
Crece en los intervalos
$$\left[- \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}, \frac{\sqrt{-4 + \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2}\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{2 x}{\left(x^{2} + 1\right)^{2}} - \frac{1}{x^{2} \log{\left(10 \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}$$
$$x_{2} = \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}$$
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[- \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}, \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}\right]$$
Convexa en los intervalos
$$\left(-\infty, - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}\right] \cup \left[\frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{2 x}{\left(x^{2} + 1\right)^{2}} - \frac{1}{x^{2} \log{\left(10 \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}$$
$$x_{2} = \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}$$
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[- \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}, \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}\right]$$
Convexa en los intervalos
$$\left(-\infty, - \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2} + \frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2}\right] \cup \left[\frac{\log{\left(10 \right)}}{2} + \frac{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}}{2} + \frac{\sqrt{- \frac{8}{3} - 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + \frac{4 \left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right) \log{\left(10 \right)}}{\sqrt{- \frac{4}{3} - \frac{2 \left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)}{3 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}}} + 2 \sqrt[3]{- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{6} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{216} + \sqrt{\frac{\left(- \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{2}}{12} - 1 + 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2}\right)^{3}}{27} + \frac{\left(- \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{2} - \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right)^{3}}{108} + \frac{\left(2 - \frac{3 \log{\left(10 \right)}^{2}}{2}\right) \left(- 4 \left(- \frac{1}{8} + \frac{3 \log{\left(10 \right)}^{2}}{64}\right) \log{\left(10 \right)}^{2} + 1\right)}{3}\right)^{2}}{4}} + \frac{\left(-1 + \frac{\log{\left(10 \right)}^{2}}{2}\right)^{2} \log{\left(10 \right)}^{2}}{4}} + \log{\left(10 \right)}^{2}}} + 2 \log{\left(10 \right)}^{2}}}{2}, \infty\right)$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(x)/log(10) - atan(x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} + \operatorname{atan}{\left(x \right)}$$
- No
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = - \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} + \operatorname{atan}{\left(x \right)}$$
- No
$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)} = - \frac{\log{\left(- x \right)}}{\log{\left(10 \right)}} - \operatorname{atan}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar