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

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

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Answer 1 · 2015-08-26T17:18:33

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: $[0,oo)$ .)

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

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Explanation:

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