Differentiable in real analysis problem

Question

Let $f$ be a real value function that is differentiable in $[a,b]$, with .

Show that there is exists $x_0$ in $(a,b)$ such that:

I don't know how to prove it. Please, could anyone give me a hint. Thanks.

Answer 1

First, it's almost certainly "show there's an $x_0 \in [a, b]$ rather than $[a. v]$.

But the hypotheses seem to have a problem. If $f(a) = f(b) = 0$ and $f$ is everywhere differentiable, then $f'(c) = 0$ for some $c \in (a, b)$ by Rolle's Theorem. Yet you've got a hypothesis that says $f'(x) \ne 0$ for all $x \in [a, b]$.

I'd carefully check that your problem is stated correctly.

As stated, the problem's easy: a false hypothesis implies any conclusion.