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Gráfico de la función y = arctg((3x^2-4x-2)/(x(x-1)(x-2)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
           /     2           \
           |  3*x  - 4*x - 2 |
f(x) = atan|-----------------|
           \x*(x - 1)*(x - 2)/
$$f{\left(x \right)} = \operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)}$$
f = atan((3*x^2 - 4*x - 2)/(((x*(x - 1))*(x - 2))))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = 2$$
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en atan((3*x^2 - 4*x - 2)/(((x*(x - 1))*(x - 2)))).
$$\operatorname{atan}{\left(\frac{-2 + \left(3 \cdot 0^{2} - 0\right)}{\left(-2\right) \left(-1\right) 0} \right)}$$
Resultado:
$$f{\left(0 \right)} = \left\langle - \frac{\pi}{2}, \frac{\pi}{2}\right\rangle$$
Punto:
(0, AccumBounds(-pi/2, pi/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
$$\frac{\frac{1}{x \left(x - 2\right) \left(x - 1\right)} \left(6 x - 4\right) + \frac{\left(- x \left(x - 1\right) - \left(x - 2\right) \left(2 x - 1\right)\right) \left(\left(3 x^{2} - 4 x\right) - 2\right)}{x^{2} \left(x - 2\right)^{2} \left(x - 1\right)^{2}}}{1 + \frac{\left(\left(3 x^{2} - 4 x\right) - 2\right)^{2}}{x^{2} \left(x - 2\right)^{2} \left(x - 1\right)^{2}}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}$$
$$x_{2} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}$$
Signos de extremos en los puntos:
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                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |               /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------                                                                |                                                                                                   | 
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                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |           /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                      /   16        /  11   \/ 57              8           |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------           /    -- + 2*3 /   -- + ------  + -------------------- |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |         /                                                                     /     9      \/    27     27             _____________          /     9      \/    27     27             _____________ |                                                                                                   | 
                    ____________________________________________________________________________________________________________________                                                                     |                                                                                                                         ____________________________________________________________________________________________________________________               __________________________________________________     |        /                                                                     /                                        /        ____          /                                        /        ____  |                                                                                                   | 
                   /             _____________                                                                                                                                                               |                                                                                                                        /             _____________                                                                                                        /             _____________                             |       /                                                                     /                                        /  11   \/ 57          /                                        /  11   \/ 57   |                                                                                                   | 
                  /             /        ____                                                                                                                                                                |                                                                                                                       /             /        ____                                                                                                        /             /        ____                              |      /                                                                     /                                    9*3 /   -- + ------        /                                    9*3 /   -- + ------  |                                                                                                   | 
                 /   32        /  11   \/ 57              8                                            88                                                                                                    |                                                                                                   14                 /   32        /  11   \/ 57              8                                            88                                           /   16        /  11   \/ 57              8                |2   \/                                                                    \/                                       \/    27     27        \/                                       \/    27     27    |                                                                                                   | 
                /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------                                                                      |                                                                                                 - -- - 2*           /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------  + 2*      /    -- + 2*3 /   -- + ------  + --------------------  + 3*|- + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------|                                                                                                   | 
               /     9      \/    27     27             _____________                __________________________________________________              __________________________________________________      |                                                                                                   3                /     9      \/    27     27             _____________                __________________________________________________           /     9      \/    27     27             _____________      \3                                                                    2                                                                                                 2                              /                                                                                                   | 
              /                                        /        ____                /             _____________                                     /             _____________                              |                                                                                                                   /                                        /        ____                /             _____________                                  /                                        /        ____                                                                                                                                                                                                                                                                                                                  | 
             /                                        /  11   \/ 57                /             /        ____                                     /             /        ____                               |                                                                                                                  /                                        /  11   \/ 57                /             /        ____                                  /                                        /  11   \/ 57                                                                                                                                                                                                                                                                                                                   | 
            /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                      /   16        /  11   \/ 57              8                 |                                                                                                                 /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                   /                                    9*3 /   -- + ------                                                                                                                                                                                                                                                                                                                  | 
           /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------           /    -- + 2*3 /   -- + ------  + --------------------       |                                                                                                                /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------       \/                                       \/    27     27                                                                                                                                                                                                                                                                                                                    | 
          /                                                                     /     9      \/    27     27             _____________          /     9      \/    27     27             _____________       |                                                                                                               /                                                                     /     9      \/    27     27             _____________                                                                                                                                                                                                                                                                                                                                                                                   | 
         /                                                                     /                                        /        ____          /                                        /        ____        |                                                                                                              /                                                                     /                                        /        ____                                                                                                                                                                                                                                                                                                                                                                                    | 
        /                                                                     /                                        /  11   \/ 57          /                                        /  11   \/ 57         |                                                                                                             /                                                                     /                                        /  11   \/ 57                                                                                                                                                                                                                                                                                                                                                                                     | 
       /                                                                     /                                    9*3 /   -- + ------        /                                    9*3 /   -- + ------        |                                                                                                            /                                                                     /                                    9*3 /   -- + ------                                                                                                                                                                                                                                                                                                                                                                                    | 
 2   \/                                                                    \/                                       \/    27     27        \/                                       \/    27     27          |                                                                                                          \/                                                                    \/                                       \/    27     27                                                                                                                                                                                                                                                                                                                                                                                      | 
(- + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------, atan|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|)
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                                                                                                                                                                                                             ||                  /   32        /  11   \/ 57              8                                            88                                                                                              | |                  /   32        /  11   \/ 57              8                                            88                                                                                              | |                /   32        /  11   \/ 57              8                                            88                                                                                              || 
                                                                                                                                                                                                             ||                 /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------                                                                | |                 /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------                                                                | |               /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------                                                                || 
                                                                                                                                                                                                             ||                /     9      \/    27     27             _____________                __________________________________________________              __________________________________________________| |                /     9      \/    27     27             _____________                __________________________________________________              __________________________________________________| |              /     9      \/    27     27             _____________                __________________________________________________              __________________________________________________|| 
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                                                                                                                                                                                                             ||              /                                        /  11   \/ 57                /             /        ____                                     /             /        ____                         | |              /                                        /  11   \/ 57                /             /        ____                                     /             /        ____                         | |            /                                        /  11   \/ 57                /             /        ____                                     /             /        ____                         || 
                                                                                                                                                                                                             ||             /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                      /   16        /  11   \/ 57              8           | |             /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                      /   16        /  11   \/ 57              8           | |           /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                      /   16        /  11   \/ 57              8           || 
                                                                                                                                                                                                             ||            /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------           /    -- + 2*3 /   -- + ------  + -------------------- | |            /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------           /    -- + 2*3 /   -- + ------  + -------------------- | |          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------           /    -- + 2*3 /   -- + ------  + -------------------- || 
                                                                                                                                                                                                             ||           /                                                                     /     9      \/    27     27             _____________          /     9      \/    27     27             _____________ | |           /                                                                     /     9      \/    27     27             _____________          /     9      \/    27     27             _____________ | |         /                                                                     /     9      \/    27     27             _____________          /     9      \/    27     27             _____________ || 
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                                                                                                                                                                                                             ||        /                                                                     /                                    9*3 /   -- + ------        /                                    9*3 /   -- + ------  | |        /                                                                     /                                    9*3 /   -- + ------        /                                    9*3 /   -- + ------  | |      /                                                                     /                                    9*3 /   -- + ------        /                                    9*3 /   -- + ------  || 
                                                                                                                                                                                                             ||  4   \/                                                                    \/                                       \/    27     27        \/                                       \/    27     27    | |  1   \/                                                                    \/                                       \/    27     27        \/                                       \/    27     27    | |2   \/                                                                    \/                                       \/    27     27        \/                                       \/    27     27    || 
                                                                                                                                                                                                             ||- - + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------|*|- - + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------|*|- + ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------|| 
                                                                                                                                                                                                             \\  3                                                                    2                                                                                                 2                              / \  3                                                                    2                                                                                                 2                              / \3                                                                    2                                                                                                 2                              // 

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                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |                                                                              /    -- - 2*3 /   -- + ------  - -------------------- + --------------------------------------------------------------- |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |              __________________________________________________             /     9      \/    27     27             _____________                __________________________________________________ |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |             /             _____________                                    /                                        /        ____                /             _____________                         |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |            /             /        ____                                    /                                        /  11   \/ 57                /             /        ____                          |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |           /   16        /  11   \/ 57              8                     /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8            |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |          /    -- + 2*3 /   -- + ------  + --------------------          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------  |                                                                                                   | 
                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                   |         /     9      \/    27     27             _____________         /                                                                     /     9      \/    27     27             _____________  |                                                                                                   | 
                                                                                   ____________________________________________________________________________________________________________________      |                                                                                                                    __________________________________________________                    ____________________________________________________________________________________________________________________     |        /                                        /        ____         /                                                                     /                                        /        ____   |                                                                                                   | 
                                                                                  /             _____________                                                                                                |                                                                                                                   /             _____________                                           /             _____________                                                                                               |       /                                        /  11   \/ 57         /                                                                     /                                        /  11   \/ 57    |                                                                                                   | 
                                                                                 /             /        ____                                                                                                 |                                                                                                                  /             /        ____                                           /             /        ____                                                                                                |      /                                    9*3 /   -- + ------       /                                                                     /                                    9*3 /   -- + ------   |                                                                                                   | 
                                                                                /   32        /  11   \/ 57              8                                            88                                     |                                                                                                   14            /   16        /  11   \/ 57              8                            /   32        /  11   \/ 57              8                                            88                                    |2   \/                                       \/    27     27       \/                                                                    \/                                       \/    27     27     |                                                                                                   | 
                                                                               /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------       |                                                                                                 - -- + 2*      /    -- + 2*3 /   -- + ------  + --------------------  + 2*           /    -- - 2*3 /   -- + ------  - -------------------- + ---------------------------------------------------------------  + 3*|- - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------|                                                                                                   | 
               __________________________________________________             /     9      \/    27     27             _____________                __________________________________________________       |                                                                                                   3           /     9      \/    27     27             _____________                /     9      \/    27     27             _____________                __________________________________________________      \3                                2                                                                                                  2                                                                 /                                                                                                   | 
              /             _____________                                    /                                        /        ____                /             _____________                               |                                                                                                              /                                        /        ____                /                                        /        ____                /             _____________                                                                                                                                                                                                                                                                                                                                         | 
             /             /        ____                                    /                                        /  11   \/ 57                /             /        ____                                |                                                                                                             /                                        /  11   \/ 57                /                                        /  11   \/ 57                /             /        ____                                                                                                                                                                                                                                                                                                                                          | 
            /   16        /  11   \/ 57              8                     /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                  |                                                                                                            /                                    9*3 /   -- + ------              /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8                                                                                                                                                                                                                                                                                                                            | 
           /    -- + 2*3 /   -- + ------  + --------------------          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------        |                                                                                                          \/                                       \/    27     27               /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------                                                                                                                                                                                                                                                                                                                  | 
          /     9      \/    27     27             _____________         /                                                                     /     9      \/    27     27             _____________        |                                                                                                                                                                                /                                                                     /     9      \/    27     27             _____________                                                                                                                                                                                                                                                                                                                  | 
         /                                        /        ____         /                                                                     /                                        /        ____         |                                                                                                                                                                               /                                                                     /                                        /        ____                                                                                                                                                                                                                                                                                                                   | 
        /                                        /  11   \/ 57         /                                                                     /                                        /  11   \/ 57          |                                                                                                                                                                              /                                                                     /                                        /  11   \/ 57                                                                                                                                                                                                                                                                                                                    | 
       /                                    9*3 /   -- + ------       /                                                                     /                                    9*3 /   -- + ------         |                                                                                                                                                                             /                                                                     /                                    9*3 /   -- + ------                                                                                                                                                                                                                                                                                                                   | 
 2   \/                                       \/    27     27       \/                                                                    \/                                       \/    27     27           |                                                                                                                                                                           \/                                                                    \/                                       \/    27     27                                                                                                                                                                                                                                                                                                                     | 
(- - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------, atan|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|)
 3                                2                                                                                                  2                                                                       |/                                                                                    ____________________________________________________________________________________________________________________\ /                                                                                    ____________________________________________________________________________________________________________________\ /                                                                                  ____________________________________________________________________________________________________________________\| 
                                                                                                                                                                                                             ||                                                                                   /             _____________                                                                                          | |                                                                                   /             _____________                                                                                          | |                                                                                 /             _____________                                                                                          || 
                                                                                                                                                                                                             ||                                                                                  /             /        ____                                                                                           | |                                                                                  /             /        ____                                                                                           | |                                                                                /             /        ____                                                                                           || 
                                                                                                                                                                                                             ||                                                                                 /   32        /  11   \/ 57              8                                            88                               | |                                                                                 /   32        /  11   \/ 57              8                                            88                               | |                                                                               /   32        /  11   \/ 57              8                                            88                               || 
                                                                                                                                                                                                             ||                                                                                /    -- - 2*3 /   -- + ------  - -------------------- + --------------------------------------------------------------- | |                                                                                /    -- - 2*3 /   -- + ------  - -------------------- + --------------------------------------------------------------- | |                                                                              /    -- - 2*3 /   -- + ------  - -------------------- + --------------------------------------------------------------- || 
                                                                                                                                                                                                             ||                __________________________________________________             /     9      \/    27     27             _____________                __________________________________________________ | |                __________________________________________________             /     9      \/    27     27             _____________                __________________________________________________ | |              __________________________________________________             /     9      \/    27     27             _____________                __________________________________________________ || 
                                                                                                                                                                                                             ||               /             _____________                                    /                                        /        ____                /             _____________                         | |               /             _____________                                    /                                        /        ____                /             _____________                         | |             /             _____________                                    /                                        /        ____                /             _____________                         || 
                                                                                                                                                                                                             ||              /             /        ____                                    /                                        /  11   \/ 57                /             /        ____                          | |              /             /        ____                                    /                                        /  11   \/ 57                /             /        ____                          | |            /             /        ____                                    /                                        /  11   \/ 57                /             /        ____                          || 
                                                                                                                                                                                                             ||             /   16        /  11   \/ 57              8                     /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8            | |             /   16        /  11   \/ 57              8                     /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8            | |           /   16        /  11   \/ 57              8                     /                                    9*3 /   -- + ------              /   16        /  11   \/ 57              8            || 
                                                                                                                                                                                                             ||            /    -- + 2*3 /   -- + ------  + --------------------          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------  | |            /    -- + 2*3 /   -- + ------  + --------------------          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------  | |          /    -- + 2*3 /   -- + ------  + --------------------          /                                       \/    27     27      27*      /    -- + 2*3 /   -- + ------  + --------------------  || 
                                                                                                                                                                                                             ||           /     9      \/    27     27             _____________         /                                                                     /     9      \/    27     27             _____________  | |           /     9      \/    27     27             _____________         /                                                                     /     9      \/    27     27             _____________  | |         /     9      \/    27     27             _____________         /                                                                     /     9      \/    27     27             _____________  || 
                                                                                                                                                                                                             ||          /                                        /        ____         /                                                                     /                                        /        ____   | |          /                                        /        ____         /                                                                     /                                        /        ____   | |        /                                        /        ____         /                                                                     /                                        /        ____   || 
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                                                                                                                                                                                                             ||        /                                    9*3 /   -- + ------       /                                                                     /                                    9*3 /   -- + ------   | |        /                                    9*3 /   -- + ------       /                                                                     /                                    9*3 /   -- + ------   | |      /                                    9*3 /   -- + ------       /                                                                     /                                    9*3 /   -- + ------   || 
                                                                                                                                                                                                             ||  4   \/                                       \/    27     27       \/                                                                    \/                                       \/    27     27     | |  1   \/                                       \/    27     27       \/                                                                    \/                                       \/    27     27     | |2   \/                                       \/    27     27       \/                                                                    \/                                       \/    27     27     || 
                                                                                                                                                                                                             ||- - - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------|*|- - - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------|*|- - ------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------|| 
                                                                                                                                                                                                             \\  3                                2                                                                                                  2                                                                 / \  3                                2                                                                                                  2                                                                 / \3                                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:
$$x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}$$
Puntos máximos de la función:
$$x_{1} = - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}$$
Decrece en los intervalos
$$\left[- \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}, - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}\right]$$
Crece en los intervalos
$$\left(-\infty, - \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2} + \frac{2}{3}\right] \cup \left[- \frac{\sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}}{2} + \frac{2}{3} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + \frac{88}{27 \sqrt{\frac{8}{9 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}}} + 2 \sqrt[3]{\frac{\sqrt{57}}{27} + \frac{11}{27}} + \frac{16}{9}}} + \frac{32}{9}}}{2}, \infty\right)$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = 2$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)} = 0$$
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 atan((3*x^2 - 4*x - 2)/(((x*(x - 1))*(x - 2)))), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)} = - \operatorname{atan}{\left(\frac{3 x^{2} + 4 x - 2}{x \left(- x - 2\right) \left(- x - 1\right)} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{\left(3 x^{2} - 4 x\right) - 2}{x \left(x - 1\right) \left(x - 2\right)} \right)} = \operatorname{atan}{\left(\frac{3 x^{2} + 4 x - 2}{x \left(- x - 2\right) \left(- x - 1\right)} \right)}$$
- No
es decir, función
no es
par ni impar