# Derivative of Monotone Function

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## Theorem

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

### Real Function with Positive Derivative is Increasing

If $\forall x \in \openint a b: \map {f'} x \ge 0$, then $f$ is increasing on $\closedint a b$.

### Real Function with Strictly Positive Derivative is Strictly Increasing

If $\forall x \in \openint a b: \map {f'} x > 0$, then $f$ is strictly increasing on $\closedint a b$.

### Real Function with Negative Derivative is Decreasing

If $\forall x \in \openint a b: \map {f'} x \le 0$, then $f$ is decreasing on $\closedint a b$.

### Real Function with Strictly Negative Derivative is Strictly Decreasing

If $\forall x \in \openint a b: \map {f'} x < 0$, then $f$ is strictly decreasing on $\closedint a b$.

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 12.7$