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-
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- Gráfico de la función y =:
-
e^3*x+10*e^2*x
-
x^11
-
(-cos(2*x)-sin(2*x))*exp(x)
-
5/(x^2-16)
-
Expresiones idénticas
- (- trece - cuatro *x^ dos + diecisiete *x)/(veinticinco - ocho *x^ dos + siete *x^ tres)
- ( menos 13 menos 4 multiplicar por x al cuadrado más 17 multiplicar por x) dividir por (25 menos 8 multiplicar por x al cuadrado más 7 multiplicar por x al cubo )
- ( menos trece menos cuatro multiplicar por x en el grado dos más diecisiete multiplicar por x) dividir por (veinticinco menos ocho multiplicar por x en el grado dos más siete multiplicar por x en el grado tres)
- (-13-4*x2+17*x)/(25-8*x2+7*x3)
- -13-4*x2+17*x/25-8*x2+7*x3
- (-13-4*x²+17*x)/(25-8*x²+7*x³)
- (-13-4*x en el grado 2+17*x)/(25-8*x en el grado 2+7*x en el grado 3)
- (-13-4x^2+17x)/(25-8x^2+7x^3)
- (-13-4x2+17x)/(25-8x2+7x3)
- -13-4x2+17x/25-8x2+7x3
- -13-4x^2+17x/25-8x^2+7x^3
- (-13-4*x^2+17*x) dividir por (25-8*x^2+7*x^3)
-
Expresiones semejantes
- (-13-4*x^2+17*x)/(25-8*x^2-7*x^3)
- (-13-4*x^2+17*x)/(25+8*x^2+7*x^3)
- (13-4*x^2+17*x)/(25-8*x^2+7*x^3)
- (-13+4*x^2+17*x)/(25-8*x^2+7*x^3)
- (-13-4*x^2-17*x)/(25-8*x^2+7*x^3)
Gráfico de la función y = (-13-4*x^2+17*x)/(25-8*x^2+7*x^3)
Solución
Ha introducido
[src]
2
-13 - 4*x + 17*x
f(x) = -----------------
2 3
25 - 8*x + 7*x
$$f{\left(x \right)} = \frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)}$$
f = (17*x - 4*x^2 - 13)/(7*x^3 + 25 - 8*x^2)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -1.22748336473913$$
$$x_{1} = -1.22748336473913$$
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:
$$\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
$$x_{2} = \frac{13}{4}$$
Solución numérica
$$x_{1} = 1$$
$$x_{2} = 3.25$$
o sea hay que resolver la ecuación:
$$\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
$$x_{2} = \frac{13}{4}$$
Solución numérica
$$x_{1} = 1$$
$$x_{2} = 3.25$$
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
$$\frac{17 - 8 x}{7 x^{3} + \left(25 - 8 x^{2}\right)} + \frac{\left(17 x + \left(- 4 x^{2} - 13\right)\right) \left(- 21 x^{2} + 16 x\right)}{\left(7 x^{3} + \left(25 - 8 x^{2}\right)\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}$$
$$x_{2} = - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}$$
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:
$$x_{1} = \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}$$
Decrece en los intervalos
$$\left(-\infty, - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}\right] \cup \left[\frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}, \infty\right)$$
Crece en los intervalos
$$\left[- \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}, \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{17 - 8 x}{7 x^{3} + \left(25 - 8 x^{2}\right)} + \frac{\left(17 x + \left(- 4 x^{2} - 13\right)\right) \left(- 21 x^{2} + 16 x\right)}{\left(7 x^{3} + \left(25 - 8 x^{2}\right)\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}$$
$$x_{2} = - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}$$
Signos de extremos en los puntos:
2
/ _____________________________________________________________________________________________________________________________________________________________________________________________________\ _____________________________________________________________________________________________________________________________________________________________________________________________________
| / ______________________________ | / ______________________________
_____________________________________________________________________________________________________________________________________________________________________________________________________ | / / ______________ | / / ______________
/ ______________________________ | / 2797 / 102400759 \/ 165018008118 18769 13107 | / 2797 / 102400759 \/ 165018008118 18769 13107
/ / ______________ | / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- | 17* / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + --------------------------------------------------------------------------------------------------------
/ 2797 / 102400759 \/ 165018008118 18769 13107 | __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ | __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________
/ ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- | / ______________________________ / / ______________ / ______________________________ | / ______________________________ / / ______________ / ______________________________
__________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ | / / ______________ / / 102400759 \/ 165018008118 / / ______________ | / / ______________ / / 102400759 \/ 165018008118 / / ______________
/ ______________________________ / / ______________ / ______________________________ | / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 | / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769
/ / ______________ / / 102400759 \/ 165018008118 / / ______________ | / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- | 17* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + -----------------------------------------
/ 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 | / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ | / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________
/ ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- | / / ______________ / / / ______________ | / / ______________ / / / ______________
/ 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ | / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 | / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118
/ / ______________ / / / ______________ | / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- | / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ----------------
/ / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 185 |17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 | \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816
/ 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- --- - 4*|-- + ---------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 8 \8 2 2 / 2 2
(-- + ---------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
8 2 2 2 3
/ _____________________________________________________________________________________________________________________________________________________________________________________________________\ / _____________________________________________________________________________________________________________________________________________________________________________________________________\
| / ______________________________ | | / ______________________________ |
| / / ______________ | | / / ______________ |
| / 2797 / 102400759 \/ 165018008118 18769 13107 | | / 2797 / 102400759 \/ 165018008118 18769 13107 |
| / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- | | / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- |
| __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ | | __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ |
| / ______________________________ / / ______________ / ______________________________ | | / ______________________________ / / ______________ / ______________________________ |
| / / ______________ / / 102400759 \/ 165018008118 / / ______________ | | / / ______________ / / 102400759 \/ 165018008118 / / ______________ |
| / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 | | / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 |
| / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- | | / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- |
| / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ | | / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ |
| / / ______________ / / / ______________ | | / / ______________ / / / ______________ |
| / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 | | / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 |
| / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- | | / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- |
|17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 | |17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 |
25 - 8*|-- + ---------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + 7*|-- + ---------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\8 2 2 / \8 2 2 / 2
/ _____________________________________________________________________________________________________________________________________________________________________________________________________\ _____________________________________________________________________________________________________________________________________________________________________________________________________
| / ______________________________ | / ______________________________
_____________________________________________________________________________________________________________________________________________________________________________________________________ | / / ______________ | / / ______________
/ ______________________________ | / 2797 / 102400759 \/ 165018008118 18769 13107 | / 2797 / 102400759 \/ 165018008118 18769 13107
/ / ______________ | / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- | 17* / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + --------------------------------------------------------------------------------------------------------
/ 2797 / 102400759 \/ 165018008118 18769 13107 | __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ | / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ __________________________________________________________________________________________
/ ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- | / ______________________________ / / ______________ / ______________________________ | / / ______________ / ______________________________ / ______________________________
__________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ | / / ______________ / / 102400759 \/ 165018008118 / / ______________ | / / 102400759 \/ 165018008118 / / ______________ / / ______________
/ ______________________________ / / ______________ / ______________________________ | / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 | / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 / 2797 / 102400759 \/ 165018008118 18769
/ / ______________ / / 102400759 \/ 165018008118 / / ______________ | / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- | / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- 17* / ---- + 2*3 / --------- + ---------------- + -----------------------------------------
/ 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 | / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ | / / 336 \/ 4741632 18816 ______________________________ / 336 \/ 4741632 18816 ______________________________
/ ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- | / / ______________ / / / ______________ | / / / ______________ / / ______________
/ 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ | / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 | / / / 102400759 \/ 165018008118 / / 102400759 \/ 165018008118
/ / ______________ / / / ______________ | / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- | / / 14112*3 / --------- + ---------------- / 14112*3 / --------- + ----------------
/ / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 185 |17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 | \/ \/ \/ 4741632 18816 \/ \/ 4741632 18816
/ 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- --- - 4*|-- + ---------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------
17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 8 \8 2 2 / 2 2
(-- + ---------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
8 2 2 2 3
/ _____________________________________________________________________________________________________________________________________________________________________________________________________\ / _____________________________________________________________________________________________________________________________________________________________________________________________________\
| / ______________________________ | | / ______________________________ |
| / / ______________ | | / / ______________ |
| / 2797 / 102400759 \/ 165018008118 18769 13107 | | / 2797 / 102400759 \/ 165018008118 18769 13107 |
| / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- | | / ---- - 2*3 / --------- + ---------------- - ----------------------------------------- + -------------------------------------------------------------------------------------------------------- |
| __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ | | __________________________________________________________________________________________ / 168 \/ 4741632 18816 ______________________________ __________________________________________________________________________________________ |
| / ______________________________ / / ______________ / ______________________________ | | / ______________________________ / / ______________ / ______________________________ |
| / / ______________ / / 102400759 \/ 165018008118 / / ______________ | | / / ______________ / / 102400759 \/ 165018008118 / / ______________ |
| / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 | | / 2797 / 102400759 \/ 165018008118 18769 / 14112*3 / --------- + ---------------- / 2797 / 102400759 \/ 165018008118 18769 |
| / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- | | / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- / \/ 4741632 18816 224* / ---- + 2*3 / --------- + ---------------- + ----------------------------------------- |
| / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ | | / 336 \/ 4741632 18816 ______________________________ / / 336 \/ 4741632 18816 ______________________________ |
| / / ______________ / / / ______________ | | / / ______________ / / / ______________ |
| / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 | | / / 102400759 \/ 165018008118 / / / 102400759 \/ 165018008118 |
| / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- | | / 14112*3 / --------- + ---------------- / / 14112*3 / --------- + ---------------- |
|17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 | |17 \/ \/ 4741632 18816 \/ \/ \/ 4741632 18816 |
25 - 8*|-- + ---------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + 7*|-- + ---------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\8 2 2 / \8 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:
$$x_{1} = \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}$$
Decrece en los intervalos
$$\left(-\infty, - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}\right] \cup \left[\frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}, \infty\right)$$
Crece en los intervalos
$$\left[- \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2} + \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8}, \frac{\sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}}{2} + \frac{17}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} - \frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + \frac{13107}{224 \sqrt{\frac{18769}{14112 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}}} + 2 \sqrt[3]{\frac{\sqrt{165018008118}}{18816} + \frac{102400759}{4741632}} + \frac{2797}{336}}} + \frac{2797}{168}}}{2}\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -1.22748336473913$$
$$x_{1} = -1.22748336473913$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-13 - 4*x^2 + 17*x)/(25 - 8*x^2 + 7*x^3), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{x \left(7 x^{3} + \left(25 - 8 x^{2}\right)\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{x \left(7 x^{3} + \left(25 - 8 x^{2}\right)\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{x \left(7 x^{3} + \left(25 - 8 x^{2}\right)\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{17 x + \left(- 4 x^{2} - 13\right)}{x \left(7 x^{3} + \left(25 - 8 x^{2}\right)\right)}\right) = 0$$
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:
$$\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)} = \frac{- 4 x^{2} - 17 x - 13}{- 7 x^{3} - 8 x^{2} + 25}$$
- No
$$\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)} = - \frac{- 4 x^{2} - 17 x - 13}{- 7 x^{3} - 8 x^{2} + 25}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)} = \frac{- 4 x^{2} - 17 x - 13}{- 7 x^{3} - 8 x^{2} + 25}$$
- No
$$\frac{17 x + \left(- 4 x^{2} - 13\right)}{7 x^{3} + \left(25 - 8 x^{2}\right)} = - \frac{- 4 x^{2} - 17 x - 13}{- 7 x^{3} - 8 x^{2} + 25}$$
- No
es decir, función
no es
par ni impar