Derivative sign implies monotonicity

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 $I$.

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 .