(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If f is differentiable on the interval [a,b] and f'(a)<0<f'(b), prove that there is a c with a<c<b for which f'(c)=0.

2. Relevant equations

3. The attempt at a solution

Well, I first tried to use IVT but I was having a hard time to I talked to my prof. and he said to use extreme value theorem and fermats theorem. So, by EVT, I know there will be an aboslute maximum and absolute minimum on [a,b]. By f'(a)<0<f'(b) I know that it will be decreasing and increasing and therefore will have a local miniimum on (a,b). Fermats theorem then says that if f(c) is a local extremum, then c must be a critical number of f. Which means the function WILL have a c to make f'(c)=0. I know this intuitively but I'm having a hard time rigorously proving it.

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# EVT and Fermats to prove f'(c)=0

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