# What are the values and types of the critical points, if any, of f(x)=(x+3x^2) / (1-x^2)?

Jul 6, 2018

There is a local maximum at $\left(- 5.83 , - 2.91\right)$ and a local minimum at $\left(- 0.17 , - 0.86\right)$.

#### Explanation:

The function is

$f \left(x\right) = \frac{x + 3 {x}^{2}}{1 - {x}^{2}}$

This function is a quotient of $2$ derivable functions

The derivative of a quotient is

$\left(\frac{u}{v}\right) ' = \frac{u ' v - u v '}{{v}^{2}}$

$u = x + 3 {x}^{2}$, $\implies$, $u ' = 1 + 6 x$

$v = 1 - {x}^{2}$, $\implies$, $u ' = - 2 x$

Therefore,

$f ' \left(x\right) = \frac{\left(1 + 6 x\right) \left(1 - {x}^{2}\right) - \left(x + 3 {x}^{2}\right) \left(- 2 x\right)}{1 - {x}^{2}} ^ 2$

$= \frac{1 - {x}^{2} + 6 x - 6 {x}^{3} + 2 {x}^{2} + 6 {x}^{3}}{1 - {x}^{2}} ^ 2$

$= \frac{{x}^{2} + 6 x + 1}{1 - {x}^{2}} ^ 2$

The critical points are when

$f ' \left(x\right) = 0$

That is

${x}^{2} + 6 x + 1 = 0$

The solutions to this quadratic equation are

$x = \frac{- 6 \pm \sqrt{36 - 4}}{2} = \frac{- 6 \pm \sqrt{32}}{2}$

$= \frac{- 6 \pm 4 \sqrt{2}}{2}$

$= - 3 \pm 2 \sqrt{2}$

Let ${x}_{1} = - 3 - 2 \sqrt{2}$

and

${x}_{2} = - 3 + 2 \sqrt{2}$

Let's build a variation chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$${x}_{1}$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a a a}$${x}_{2}$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a}$↗$\textcolor{w h i t e}{a a}$color(white)(aaa)↘$\textcolor{w h i t e}{a a a}$↘$\textcolor{w h i t e}{a a a}$color(white)(aa)↗$\textcolor{w h i t e}{a a a}$↗

Therefore,

There is a local maximum at $\left(- 5.83 , - 2.91\right)$ and a local minimum at $\left(- 0.17 , - 0.86\right)$.

graph{(x+3x^2)/(1-x^2) [-10, 10, -5, 5]}