$f'(x_0)=0?$ See below.

Question

Given $f:RR->RR$ differentiable in $RR$ .
Given $a,b$ with $a<$ $b$ for which $f'(a)<0$ and $f'(b)>0$ . Prove that there is $x_0$ $in$ $(a,b)$ for which $f'(x_0)=0$

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Answer 1 · 2018-03-03T22:05:39

Here is a sketch.

$f$ is continuous on $[a,b]$ , so $f$ has a minimum of $[a,b]$ .

That minimum cannot be at $a$ or $b$ .

The minimum occurs at a relative minimum, and be Fermat's Theorem, the derivative at that point is $0$ .