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

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#

1 Answer
Mar 3, 2018

Here is a sketch.

Explanation:

#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#.