# Find the intervals of increase and /or decrease of f(x) = X^2e^2 and determine all local max and min points if any?

May 26, 2018

$f$ is decreasing in $\left(- \infty , 0\right]$, increasing in $\left[0 , + \infty\right)$ and has a global and so local minimum at $x = 0$, $f \left(0\right) = 0$

#### Explanation:

$f \left(x\right) = {e}^{2} {x}^{2}$

graph{e^2x^2 [-5.095, 4.77, -1.34, 3.59]}

The domain of $f$ is $\mathbb{R}$

Notice that $f \left(0\right) = 0$

Now, $f ' \left(x\right) = 2 {e}^{2} x$

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

Variance table

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a 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 a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a 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 a a a a a}$↘$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$↗

So $f$ is decreasing in $\left(- \infty , 0\right]$, increasing in $\left[0 , + \infty\right)$ and has a global and so local minimum at $x = 0$, $f \left(0\right) = 0$
We also get $f \left(x\right) \ge 0$, $\forall$$x$$\in$$\mathbb{R}$