> [!NOTE] Corollary (Functions with positive derivative) > If $f:I\to \mathbb{R}$ is [[Differentiable real functions|differentiable]] on the [[Real intervals|open interval]] $I$ and $f'(x)>0$ for all $x\in I$ then $f$ is strictly increasing on the interval. ***Proof***. If there are two points $a$ and $b$ with $a<b$ but $f(a)\geq f(b)$ then we could find a point $c$ where $f'(c) = \frac{f(b)-f(a)}{b-a} \leq 0$contradicting the hypothesis.