Question

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

Answer 1

`f` is decreasing in `(-oo,0]`, increasing in `[0,+oo)` and has a global and so local minimum at `x=0`, `f(0)=0`

Explanation:

`f(x)=e^2x^2`

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

The domain of `f` is `RR`

Notice that `f(0)=0`

Now, `f'(x)=2e^2x`

`f'(0)=0`

Variance table

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaaaaaaaaa)` `0` `color(white)(aaaaaaaaaa)` `+oo`

`color(white)(aaaa)` `f'(x)` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaaaa)` `0` `color(white)(aaaaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaa)` `↘` `color(white)(aaaaaa)` `0` `color(white)(aaaaaa)` `↗`

So `f` is decreasing in `(-oo,0]`, increasing in `[0,+oo)` and has a global and so local minimum at `x=0`, `f(0)=0`
We also get `f(x)>=0`, `AA` `x` `in` `RR`