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

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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#

Given

Given

##### 1 Answer

Mar 3, 2018

Here is a sketch.

#### Explanation:

That minimum cannot be at

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