Question

How do you find all intervals where the function `f(x)=e^(x^2)` is increasing?

Answer 1

Investigate the sign of `f'(x)`.

Explanation:

On intervals on which `f'(x)` is positive (`>0`), `f(x)` is increasing.

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

`f'(x)=2xe^(x^2)`

Because `f'(x)` is never undefined, it could possibly change sign only at `x` values where `f'(x) =0`

`2xe^(x^2) = 0` if and only if

`2x = 0`, so `x = 0` #

or `e^(x^2) = 0` but `e^n` is never `0` for any `n`.

Since `2e^(x^2) > 0` for all `x`, the sign of `f'(x)` is the same as the sign of `x`

which is (of course) negative for `x<0` and positive for `x>0`.

So `f` is increasing on the interval `(0, oo)`.

(In my experience the usual practice is to state open intervals on which a function is increasing. It is also true that this function is increasing on the closed interval: `[0,oo)`.)