Upper limit of differential connection to original function

Question

I have a homework with a question that goes like this:

Let $f(x)$ be a differentiable function at $[a,b]$. assuming:

1. $f(a)=f(b)=0$
2. $f(x)>0$ for all $x$ in $(a,b)$
3. there is a M that: $|f^′(x)|\le M$ for every $x$ in $(a,b)$

For every $x$ in $(a,b)$ prove that $$2f(x)\le M(b-a).$$

I know it has to do something with Rolle's theorem but no idea how to approach it.

What is the right way to approach questions like this and what signs do I have to know which direction to take it ? I have an upper limit so therefore I need to prove it by assuming something wrong. Yet I don't know how should I choose my $M$ so that I can disprove a wrong assumption.