Monotonicity compares function values as the input moves from left to right. The interval and differ

Question

Monotonicity compares function values as the input moves from left to right. The interval and differentiability assumptions matter.

Increasing behaviour

On an interval where $\mathrm{f}$ is differentiable, $\mathrm{f}'(x)\geq0$ characterises a non-decreasing function. The stronger condition $\mathrm{f}'(x)>0$ is sufficient, but not necessary, for strict increase.

$\mathrm{f}(x)=e^x$

A non-decreasing step function

$\mathrm{h}(x)=x^3$

The step function has derivative zero on each open constant piece but is not differentiable at its jumps. The function $x^3$ is strictly increasing even though its derivative is zero at $x=0$.

Decreasing behaviour

On an interval where $\mathrm{f}$ is differentiable, $\mathrm{f}'(x)\leq0$ characterises a non-increasing function. The stronger condition $\mathrm{f}'(x)<0$ is sufficient, but not necessary, for strict decrease.

A local maximum or minimum is defined relative to a neighbourhood of the point.

First-derivative test

Local maximum

$x$	$a^-$	$a$	$a^+$
$\mathrm{f}'(x)$	Positive	0	Negative

Local minimum

$x$	$a^-$	$a$	$a^+$
$\mathrm{f}'(x)$	Negative	0	Positive

A stationary point satisfies $f^{\prime}(a)=0$. If the first derivative has the same sign on both sides, the point is not a local extremum. An inflection additionally requires a change of concavity; $f^{\prime\prime}(a)=0$ alone is inconclusive.

Accepted Answer

Maximum: $\mathrm f'$ changes $+$ to $-$. Minimum: $\mathrm f'$ changes $-$ to $+$. Stationary inflexion: $\mathrm f'(a)=0$, $\mathrm f'$ keeps the same sign, and concavity changes.

First-Derivative Tests and Monotonicity

First-Derivative Tests and Monotonicity

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