Question

`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`

Answer 1

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`.