Derivative of Monotone Function

Theorem

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

Let:

$\forall x \in \openint a b: \map {f'} x \ge 0$

Then $f$ is increasing on $\closedint a b$.

Let:

$\forall x \in \openint a b: \map {f'} x > 0$

Then $f$ is strictly increasing on $\closedint a b$.

Let:

$\forall x \in \openint a b: \map {f'} x \le 0$

Then $f$ is decreasing on $\closedint a b$.

Let:

$\forall x \in \openint a b: \map {f'} x < 0$

Then $f$ is strictly decreasing on $\closedint a b$.

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