Let $a,b in mathbb{R}$ with $a < b$ and $f : (a,b) to mathbb{R}$ be differentiable with

$$

f(a) leq 0, quad f(b) < 0 quad text{and} quad int_{a}^b f(x)text{dx} geq 0.

$$

I have to show that there exists a $x_0 in (a,b)$ with $f'(x_0) = 0$.

Here is my attempt: From the mean-value theorem I know that there exists a point $c in (a,b)$ such that

$$

f'(c) = frac{f(b) – f(a)}{b-a}

$$

and from the mean value theorem for definite integrals I know that there exists some point $xi in (a,b)$ such that

$$

int_{a}^b f(x)text{dx} = f(xi)(b-a).

$$

From here I don’t know what to do next. Can you give me a tip how I can continue? The case $f (a) = f (b)$ is clear, so that we can assume that $f (a) neq f(b)$.