Derivative sign implies monotonicity - Imported Knowlpedia Content

Derivative sign implies monotonicity: Let $I\subseteq\mathbb{R}$ be an interval, and let $f:I\to\mathbb{R}$ be differentiable on $I$ .

If $f'(x)\ge 0$ for all $x\in I$ , then $f$ is nondecreasing (monotone increasing) on $I$ .
If $f'(x)\le 0$ for all $x\in I$ , then $f$ is nonincreasing (monotone decreasing) on $I$ .

Remarks

This is proved by applying the mean value theorem on subintervals and interpreting the sign of the derivative as controlling slopes. The strict version is positive derivative implies increasing, and the conclusion is a special case of being a monotone function.