Starting from the theorem that any continuous function on a closed and bounded interval attains a maximum value, prove Rolle's Theorem. Deduce the Mean Value Theorem.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. If $f^{\prime}(t)>0$ for all $t$ show that $f$ is a strictly increasing function.

Conversely, if $f$ is strictly increasing, is $f^{\prime}(t)>0$ for all $t$ ?