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- Gráfico de la función implícita
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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
y=(x+1)^3
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y=2x
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2*x^3-3*x
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3-x^2
-
Expresiones idénticas
- ((x- uno)(x- dos)*x)/((x- ocho)(x- siete))
- ((x menos 1)(x menos 2) multiplicar por x) dividir por ((x menos 8)(x menos 7))
- ((x menos uno)(x menos dos) multiplicar por x) dividir por ((x menos ocho)(x menos siete))
- ((x-1)(x-2)x)/((x-8)(x-7))
- x-1x-2x/x-8x-7
- ((x-1)(x-2)*x) dividir por ((x-8)(x-7))
-
Expresiones semejantes
- ((x-1)(x+2)*x)/((x-8)(x-7))
- ((x-1)(x-2)*x)/((x+8)(x-7))
- ((x-1)(x-2)*x)/((x-8)(x+7))
- ((x+1)(x-2)*x)/((x-8)(x-7))
Gráfico de la función y = ((x-1)(x-2)*x)/((x-8)(x-7))
Solución
Ha introducido
[src]
(x - 1)*(x - 2)*x
f(x) = -----------------
(x - 8)*(x - 7)
$$f{\left(x \right)} = \frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)}$$
f = (x*((x - 2)*(x - 1)))/(((x - 8)*(x - 7)))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 7$$
$$x_{2} = 8$$
$$x_{1} = 7$$
$$x_{2} = 8$$
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:
$$\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = 2$$
Solución numérica
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = 2$$
o sea hay que resolver la ecuación:
$$\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = 2$$
Solución numérica
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = 2$$
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
$$\frac{x \left(15 - 2 x\right) \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right)^{2} \left(x - 7\right)^{2}} + \frac{1}{\left(x - 8\right) \left(x - 7\right)} \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = \frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}$$
$$x_{2} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
$$x_{3} = \frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
$$x_{4} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
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:
$$x_{1} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
$$x_{2} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
Puntos máximos de la función:
$$x_{2} = \frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}$$
$$x_{2} = \frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
Decrece en los intervalos
$$\left(-\infty, \frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}\right] \cap \left(-\infty, \frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}\right] \cap \left[\frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}, \infty\right) \cap \left[\frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}\right] \cap \left(-\infty, \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}\right] \cap \left[\frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}, \infty\right) \cap \left[\frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}, \infty\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
$$\frac{x \left(15 - 2 x\right) \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right)^{2} \left(x - 7\right)^{2}} + \frac{1}{\left(x - 8\right) \left(x - 7\right)} \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = \frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}$$
$$x_{2} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
$$x_{3} = \frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
$$x_{4} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
Signos de extremos en los puntos:
/ _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\
| / __________________________ | | / __________________________ | | / __________________________ |
_________________________________________________________________________________________________________________________________________________________________________ | / / _______ | | / / _______ | | / / _______ |
/ __________________________ | / 506 1092 / 1857115 35*I*\/ 57606 15625 | | / 506 1092 / 1857115 35*I*\/ 57606 15625 | | / 506 1092 / 1857115 35*I*\/ 57606 15625 |
/ / _______ | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- |
/ 506 1092 / 1857115 35*I*\/ 57606 15625 | ______________________________________________________________________________ / 3 ______________________________________________________________________________ \/ 216 3 __________________________ | | ______________________________________________________________________________ / 3 ______________________________________________________________________________ \/ 216 3 __________________________ | | ______________________________________________________________________________ / 3 ______________________________________________________________________________ \/ 216 3 __________________________ |
/ --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ |
______________________________________________________________________________ / 3 ______________________________________________________________________________ \/ 216 3 __________________________ | / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 |
/ __________________________ / / __________________________ / _______ | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- |
/ / _______ / / / _______ / 1857115 35*I*\/ 57606 | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 |
/ 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ |
/ --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | / / _______ / / / _______ | | / / _______ / / / _______ | | / / _______ / / / _______ |
/ 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |
/ / _______ / / / _______ | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |
/ / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |11 \/ \/ 216 3 \/ \/ \/ 216 3 | |13 \/ \/ 216 3 \/ \/ \/ 216 3 | |15 \/ \/ 216 3 \/ \/ \/ 216 3 |
/ 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |-- - ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- - ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- - ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
15 \/ \/ 216 3 \/ \/ \/ 216 3 \2 2 2 / \2 2 2 / \2 2 2 /
(-- - ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
2 2 2 / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\
| / __________________________ | | / __________________________ |
| / / _______ | | / / _______ |
| / 506 1092 / 1857115 35*I*\/ 57606 15625 | | / 506 1092 / 1857115 35*I*\/ 57606 15625 |
| / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- |
| ______________________________________________________________________________ / 3 ______________________________________________________________________________ \/ 216 3 __________________________ | | ______________________________________________________________________________ / 3 ______________________________________________________________________________ \/ 216 3 __________________________ |
| / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ |
| / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 |
| / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- |
| / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 |
| / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ |
| / / _______ / / / _______ | | / / _______ / / / _______ |
| / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |
| / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |
|1 \/ \/ 216 3 \/ \/ \/ 216 3 | | 1 \/ \/ 216 3 \/ \/ \/ 216 3 |
|- - ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|- - - ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\2 2 2 / \ 2 2 2 / / _________________________________________________________________________________________________________________________________________________________________________ \ / _________________________________________________________________________________________________________________________________________________________________________ \ / _________________________________________________________________________________________________________________________________________________________________________ \
| / __________________________ | | / __________________________ | | / __________________________ |
_________________________________________________________________________________________________________________________________________________________________________ | / / _______ | | / / _______ | | / / _______ |
/ __________________________ | / 506 1092 / 1857115 35*I*\/ 57606 15625 | | / 506 1092 / 1857115 35*I*\/ 57606 15625 | | / 506 1092 / 1857115 35*I*\/ 57606 15625 |
/ / _______ | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- |
/ 506 1092 / 1857115 35*I*\/ 57606 15625 | / 3 ______________________________________________________________________________ \/ 216 3 __________________________ ______________________________________________________________________________| | / 3 ______________________________________________________________________________ \/ 216 3 __________________________ ______________________________________________________________________________| | / 3 ______________________________________________________________________________ \/ 216 3 __________________________ ______________________________________________________________________________|
/ --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | / / __________________________ / _______ / __________________________ | | / / __________________________ / _______ / __________________________ | | / / __________________________ / _______ / __________________________ |
/ 3 ______________________________________________________________________________ \/ 216 3 __________________________ ______________________________________________________________________________ | / / / _______ / 1857115 35*I*\/ 57606 / / _______ | | / / / _______ / 1857115 35*I*\/ 57606 / / _______ | | / / / _______ / 1857115 35*I*\/ 57606 / / _______ |
/ / __________________________ / _______ / __________________________ | / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- / 253 / 1857115 35*I*\/ 57606 15625 | | / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- / 253 / 1857115 35*I*\/ 57606 15625 | | / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- / 253 / 1857115 35*I*\/ 57606 15625 |
/ / / _______ / 1857115 35*I*\/ 57606 / / _______ | / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 / --- + 2*3 / ------- + -------------- + ---------------------------------- | | / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 / --- + 2*3 / ------- + -------------- + ---------------------------------- | | / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 / --- + 2*3 / ------- + -------------- + ---------------------------------- |
/ / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- / 253 / 1857115 35*I*\/ 57606 15625 | / / 3 \/ 216 3 __________________________ / 3 \/ 216 3 __________________________ | | / / 3 \/ 216 3 __________________________ / 3 \/ 216 3 __________________________ | | / / 3 \/ 216 3 __________________________ / 3 \/ 216 3 __________________________ |
/ / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 / --- + 2*3 / ------- + -------------- + ---------------------------------- | / / / _______ / / _______ | | / / / _______ / / _______ | | / / / _______ / / _______ |
/ / 3 \/ 216 3 __________________________ / 3 \/ 216 3 __________________________ | / / / 1857115 35*I*\/ 57606 / / 1857115 35*I*\/ 57606 | | / / / 1857115 35*I*\/ 57606 / / 1857115 35*I*\/ 57606 | | / / / 1857115 35*I*\/ 57606 / / 1857115 35*I*\/ 57606 |
/ / / _______ / / _______ | / / 18*3 / ------- + -------------- / 18*3 / ------- + -------------- | | / / 18*3 / ------- + -------------- / 18*3 / ------- + -------------- | | / / 18*3 / ------- + -------------- / 18*3 / ------- + -------------- |
/ / / 1857115 35*I*\/ 57606 / / 1857115 35*I*\/ 57606 |11 \/ \/ \/ 216 3 \/ \/ 216 3 | |13 \/ \/ \/ 216 3 \/ \/ 216 3 | |15 \/ \/ \/ 216 3 \/ \/ 216 3 |
/ / 18*3 / ------- + -------------- / 18*3 / ------- + -------------- |-- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|*|-- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|*|-- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|
15 \/ \/ \/ 216 3 \/ \/ 216 3 \2 2 2 / \2 2 2 / \2 2 2 /
(-- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
2 2 2 / _________________________________________________________________________________________________________________________________________________________________________ \ / _________________________________________________________________________________________________________________________________________________________________________ \
| / __________________________ | | / __________________________ |
| / / _______ | | / / _______ |
| / 506 1092 / 1857115 35*I*\/ 57606 15625 | | / 506 1092 / 1857115 35*I*\/ 57606 15625 |
| / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- | | / --- - ---------------------------------------------------------------------------------------- - 2*3 / ------- + -------------- - ---------------------------------- |
| / 3 ______________________________________________________________________________ \/ 216 3 __________________________ ______________________________________________________________________________| | / 3 ______________________________________________________________________________ \/ 216 3 __________________________ ______________________________________________________________________________|
| / / __________________________ / _______ / __________________________ | | / / __________________________ / _______ / __________________________ |
| / / / _______ / 1857115 35*I*\/ 57606 / / _______ | | / / / _______ / 1857115 35*I*\/ 57606 / / _______ |
| / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- / 253 / 1857115 35*I*\/ 57606 15625 | | / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- / 253 / 1857115 35*I*\/ 57606 15625 |
| / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 / --- + 2*3 / ------- + -------------- + ---------------------------------- | | / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 / --- + 2*3 / ------- + -------------- + ---------------------------------- |
| / / 3 \/ 216 3 __________________________ / 3 \/ 216 3 __________________________ | | / / 3 \/ 216 3 __________________________ / 3 \/ 216 3 __________________________ |
| / / / _______ / / _______ | | / / / _______ / / _______ |
| / / / 1857115 35*I*\/ 57606 / / 1857115 35*I*\/ 57606 | | / / / 1857115 35*I*\/ 57606 / / 1857115 35*I*\/ 57606 |
| / / 18*3 / ------- + -------------- / 18*3 / ------- + -------------- | | / / 18*3 / ------- + -------------- / 18*3 / ------- + -------------- |
|1 \/ \/ \/ 216 3 \/ \/ 216 3 | | 1 \/ \/ \/ 216 3 \/ \/ 216 3 |
|- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|*|- - + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|
\2 2 2 / \ 2 2 2 / / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\
| / __________________________ | | / __________________________ | | / __________________________ |
_________________________________________________________________________________________________________________________________________________________________________ | / / _______ | | / / _______ | | / / _______ |
/ __________________________ | / 506 / 1857115 35*I*\/ 57606 1092 15625 | | / 506 / 1857115 35*I*\/ 57606 1092 15625 | | / 506 / 1857115 35*I*\/ 57606 1092 15625 |
/ / _______ | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- |
/ 506 / 1857115 35*I*\/ 57606 1092 15625 | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ |
/ --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ |
______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 |
/ __________________________ / / __________________________ / _______ | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- |
/ / _______ / / / _______ / 1857115 35*I*\/ 57606 | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 |
/ 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ |
/ --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | / / _______ / / / _______ | | / / _______ / / / _______ | | / / _______ / / / _______ |
/ 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |
/ / _______ / / / _______ | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |
/ / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |11 \/ \/ 216 3 \/ \/ \/ 216 3 | |13 \/ \/ 216 3 \/ \/ \/ 216 3 | |15 \/ \/ 216 3 \/ \/ \/ 216 3 |
/ 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |-- + ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- + ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- + ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
15 \/ \/ 216 3 \/ \/ \/ 216 3 \2 2 2 / \2 2 2 / \2 2 2 /
(-- + ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
2 2 2 / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\
| / __________________________ | | / __________________________ |
| / / _______ | | / / _______ |
| / 506 / 1857115 35*I*\/ 57606 1092 15625 | | / 506 / 1857115 35*I*\/ 57606 1092 15625 |
| / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- |
| ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ |
| / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ |
| / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 |
| / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- |
| / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 |
| / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ |
| / / _______ / / / _______ | | / / _______ / / / _______ |
| / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |
| / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |
|1 \/ \/ 216 3 \/ \/ \/ 216 3 | | 1 \/ \/ 216 3 \/ \/ \/ 216 3 |
|- + ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|- - + ---------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\2 2 2 / \ 2 2 2 / / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\
| / __________________________ | | / __________________________ | | / __________________________ |
_________________________________________________________________________________________________________________________________________________________________________ | / / _______ | | / / _______ | | / / _______ |
/ __________________________ | / 506 / 1857115 35*I*\/ 57606 1092 15625 | | / 506 / 1857115 35*I*\/ 57606 1092 15625 | | / 506 / 1857115 35*I*\/ 57606 1092 15625 |
/ / _______ | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- |
/ 506 / 1857115 35*I*\/ 57606 1092 15625 | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ |
/ --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ |
______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 |
/ __________________________ / / __________________________ / _______ | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- |
/ / _______ / / / _______ / 1857115 35*I*\/ 57606 | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 |
/ 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ |
/ --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | / / _______ / / / _______ | | / / _______ / / / _______ | | / / _______ / / / _______ |
/ 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |
/ / _______ / / / _______ | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |
/ / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |11 \/ \/ 216 3 \/ \/ \/ 216 3 | |13 \/ \/ 216 3 \/ \/ \/ 216 3 | |15 \/ \/ 216 3 \/ \/ \/ 216 3 |
/ 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |-- + ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- + ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- + ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
15 \/ \/ 216 3 \/ \/ \/ 216 3 \2 2 2 / \2 2 2 / \2 2 2 /
(-- + ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
2 2 2 / _________________________________________________________________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________________________________________________________\
| / __________________________ | | / __________________________ |
| / / _______ | | / / _______ |
| / 506 / 1857115 35*I*\/ 57606 1092 15625 | | / 506 / 1857115 35*I*\/ 57606 1092 15625 |
| / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- | | / --- - 2*3 / ------- + -------------- + ---------------------------------------------------------------------------------------- - ---------------------------------- |
| ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ | | ______________________________________________________________________________ / 3 \/ 216 3 ______________________________________________________________________________ __________________________ |
| / __________________________ / / __________________________ / _______ | | / __________________________ / / __________________________ / _______ |
| / / _______ / / / _______ / 1857115 35*I*\/ 57606 | | / / _______ / / / _______ / 1857115 35*I*\/ 57606 |
| / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- | | / 253 / 1857115 35*I*\/ 57606 15625 / / 253 / 1857115 35*I*\/ 57606 15625 18*3 / ------- + -------------- |
| / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 | | / --- + 2*3 / ------- + -------------- + ---------------------------------- / / --- + 2*3 / ------- + -------------- + ---------------------------------- \/ 216 3 |
| / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ | | / 3 \/ 216 3 __________________________ / / 3 \/ 216 3 __________________________ |
| / / _______ / / / _______ | | / / _______ / / / _______ |
| / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 | | / / 1857115 35*I*\/ 57606 / / / 1857115 35*I*\/ 57606 |
| / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- | | / 18*3 / ------- + -------------- / / 18*3 / ------- + -------------- |
|1 \/ \/ 216 3 \/ \/ \/ 216 3 | | 1 \/ \/ 216 3 \/ \/ \/ 216 3 |
|- + ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|- - + ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\2 2 2 / \ 2 2 2 / 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:
$$x_{1} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
$$x_{2} = \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
Puntos máximos de la función:
$$x_{2} = \frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}$$
$$x_{2} = \frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}$$
Decrece en los intervalos
$$\left(-\infty, \frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}\right] \cap \left(-\infty, \frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}\right] \cap \left[\frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}, \infty\right) \cap \left[\frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}\right] \cap \left(-\infty, \frac{15}{2} + \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}\right] \cap \left[\frac{15}{2} - \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} - \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2}, \infty\right) \cap \left[\frac{15}{2} - \frac{\sqrt{\frac{506}{3} - 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}} + \frac{1092}{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}} - \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}}{2} + \frac{\sqrt{\frac{253}{3} + \frac{15625}{18 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}} + 2 \sqrt[3]{\frac{1857115}{216} + \frac{35 \sqrt{57606} i}{3}}}}{2}, \infty\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
$$\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{4 \sqrt[3]{5}}{3} - \frac{2 \cdot 5^{\frac{2}{3}}}{3} + \frac{16}{3}$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 7$$
$$x_{2} = 8$$
$$\lim_{x \to 7^-}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = \infty$$
$$\lim_{x \to 7^+}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{1} = 7$$
- es el punto de flexión
$$\lim_{x \to 8^-}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = -\infty$$
$$\lim_{x \to 8^+}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = \infty$$
- los límites no son iguales, signo
$$x_{2} = 8$$
- es el punto de flexión
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
$$\left[- \frac{4 \sqrt[3]{5}}{3} - \frac{2 \cdot 5^{\frac{2}{3}}}{3} + \frac{16}{3}, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, - \frac{4 \sqrt[3]{5}}{3} - \frac{2 \cdot 5^{\frac{2}{3}}}{3} + \frac{16}{3}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{4 \sqrt[3]{5}}{3} - \frac{2 \cdot 5^{\frac{2}{3}}}{3} + \frac{16}{3}$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 7$$
$$x_{2} = 8$$
$$\lim_{x \to 7^-}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = \infty$$
$$\lim_{x \to 7^+}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = -\infty$$
- los límites no son iguales, signo
$$x_{1} = 7$$
- es el punto de flexión
$$\lim_{x \to 8^-}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = -\infty$$
$$\lim_{x \to 8^+}\left(\frac{6 x + \frac{x \left(x - 2\right) \left(x - 1\right) \left(\left(2 x - 15\right) \left(\frac{1}{x - 7} + \frac{1}{x - 8}\right) - 2 + \frac{2 x - 15}{x - 7} + \frac{2 x - 15}{x - 8}\right)}{\left(x - 8\right) \left(x - 7\right)} - 6 - \frac{2 \left(2 x - 15\right) \left(x \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)}{\left(x - 8\right) \left(x - 7\right)}}{\left(x - 8\right) \left(x - 7\right)}\right) = \infty$$
- los límites no son iguales, signo
$$x_{2} = 8$$
- es el punto de flexión
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
$$\left[- \frac{4 \sqrt[3]{5}}{3} - \frac{2 \cdot 5^{\frac{2}{3}}}{3} + \frac{16}{3}, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, - \frac{4 \sqrt[3]{5}}{3} - \frac{2 \cdot 5^{\frac{2}{3}}}{3} + \frac{16}{3}\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 7$$
$$x_{2} = 8$$
$$x_{1} = 7$$
$$x_{2} = 8$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{x \to -\infty}\left(\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)}\right) = \infty$$
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 (((x - 1)*(x - 2))*x)/(((x - 8)*(x - 7))), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{1}{\left(x - 8\right) \left(x - 7\right)} \left(x - 2\right) \left(x - 1\right)\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{1}{\left(x - 8\right) \left(x - 7\right)} \left(x - 2\right) \left(x - 1\right)\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{1}{\left(x - 8\right) \left(x - 7\right)} \left(x - 2\right) \left(x - 1\right)\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{1}{\left(x - 8\right) \left(x - 7\right)} \left(x - 2\right) \left(x - 1\right)\right) = 1$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x$$
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:
$$\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)} = - \frac{x \left(- x - 2\right) \left(- x - 1\right)}{\left(- x - 8\right) \left(- x - 7\right)}$$
- No
$$\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)} = \frac{x \left(- x - 2\right) \left(- x - 1\right)}{\left(- x - 8\right) \left(- x - 7\right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)} = - \frac{x \left(- x - 2\right) \left(- x - 1\right)}{\left(- x - 8\right) \left(- x - 7\right)}$$
- No
$$\frac{x \left(x - 2\right) \left(x - 1\right)}{\left(x - 8\right) \left(x - 7\right)} = \frac{x \left(- x - 2\right) \left(- x - 1\right)}{\left(- x - 8\right) \left(- x - 7\right)}$$
- No
es decir, función
no es
par ni impar