Let $f:[a,b]\rightarrow \mathbb{R}$ a continuous function, with continuous derivate in $[a,b]$ such that: $0<f'(x)<M \ \ \forall \ x\in[a,b]$
Find $c,d\in R$ such that $c\leq f(x)\leq d$.

If you know a bit of general topology, there is an easy proof of a stronger statement:
Lemma. Let $(X, \tau), (Y, \sigma)$ be topological spaces and let $f \colon (X, \tau) \to (Y, \sigma)$ be continuous. If $C$ is compact in $(X,\tau)$. Then $f[C]$ is compact in $(Y, \sigma)$.
Prove this and combine it with the fact that $C \subseteq \mathbb{R}$ is compact in $\mathbb{R}$ if and only if it is closed and bounded.
