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

Aug 26, 2015

Investigate the sign of $f ' \left(x\right)$.

#### Explanation:

On intervals on which $f ' \left(x\right)$ is positive ($> 0$), $f \left(x\right)$ is increasing.

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

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

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

$2 x {e}^{{x}^{2}} = 0$ if and only if

$2 x = 0$, so $x = 0$ #

or ${e}^{{x}^{2}} = 0$ but ${e}^{n}$ is never $0$ for any $n$.

Since $2 {e}^{{x}^{2}} > 0$ for all $x$, the sign of $f ' \left(x\right)$ 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 $\left(0 , \infty\right)$.

(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: $\left[0 , \infty\right)$.)