Properties of a $C^1$-function $f: [0,1] \to \mathbb{R}$

Assume we have a $C^1$-function $f: [0,1] \to \mathbb{R}$ with the following properties:

• $f(0)<0$ and $f(1)>0$

I was asking myself if it is the case that there exists a point $x_0 \in (0,1)$with $f(x_0)<0$ and $f'(x_0)>0$.I‘m pretty sure that such a point has to exist, but all my efforts to proof it failed at some point.Thanks in advance.