Derivative of Monotone Function

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,. \, . \, b}\right]$ and differentiable on the open interval $\left({a \, . \, . \, b}\right)$.


 * If $\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) \ge 0$, then $f$ is increasing on $\left[{a \, . \, . \, b}\right]$.
 * If $\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) > 0$, then $f$ is strictly increasing on $\left[{a \, . \, . \, b}\right]$.


 * If $\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) \le 0$, then $f$ is decreasing on $\left[{a \, . \, . \, b}\right]$.
 * If $\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) < 0$, then $f$ is strictly decreasing on $\left[{a \, . \, . \, b}\right]$.

Proof
Let $c, d \in \left[{a \,. \, . \, b}\right]: c < d$.

Then $f$ satisfies the conditions of the Mean Value Theorem on $\left[{c \,. \, . \, d}\right]$.

Hence $\exists \xi \in \left({c \, . \, . \, d}\right): f^{\prime} \left({\xi}\right) = \frac {f \left({d}\right) - f \left({c}\right)} {d - c}$.

If $\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) \ge 0$, then $f^{\prime} \left({\xi}\right) \ge 0$ and hence $f \left({d}\right) \ge f \left({c}\right)$.

Thus $f$ is increasing on $\left[{a \,. \, . \, b}\right]$

If $\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) > 0$, then $f^{\prime} \left({\xi}\right) > 0$ and hence $f \left({d}\right) > f \left({c}\right)$.

Thus $f$ is strictly increasing on $\left[{a \,. \, . \, b}\right]$.

The other cases follow similarly.