A local maximum or minimum is defined relative to a neighbourhood of the point. Second-derivative te

Question

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

Second-derivative test

Conditions at $x=a$	Conclusion	How the gradient changes
$\mathrm{f}'(a)=0$, $\mathrm{f}''(a)<0$	Local maximum	$\mathrm{f}'$ is decreasing through zero
$\mathrm{f}'(a)=0$, $\mathrm{f}''(a)>0$	Local minimum	$\mathrm{f}'$ is increasing through zero

The second derivative is the rate of change of the gradient: $\mathrm{f}''(x)=\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(\mathrm{f}'(x)\bigr)$.

An inflection point is a point where the curve changes concavity: from concave up to concave down, or from concave down to concave up.

Non-stationary inflection points

Stationary points of inflection

The equation $\mathrm{f}''(a)=0$, or non-existence of $\mathrm{f}''(a)$, identifies only a candidate. Confirm that concavity—equivalently, the sign of $\mathrm{f}''$—changes across $a$.

A stationary point of inflection also satisfies $\mathrm{f}'(a)=0$. Not every inflection point is stationary.

First-derivative tests for stationary points of inflection

$x$	$a^-$	$a$	$a^+$
$\mathrm{f}'(x)$	Negative	0	Negative
$\mathrm{f}''$	Positive to zero to negative: concavity changes from up to down.

$x$	$a^-$	$a$	$a^+$
$\mathrm{f}'(x)$	Positive	0	Positive
$\mathrm{f}''$	Negative to zero to positive: concavity changes from down to up.

In both cases $\mathrm{f}'$ does not change sign, so the point is not a local maximum or minimum. The change of concavity establishes the inflection.

Accepted Answer

At $\mathrm f'(a)=0$: $\mathrm f''(a)0$ gives a local minimum, and $\mathrm f''(a)=0$ is inconclusive. A stationary inflexion requires a change in concavity.

Conditions at \(x=a\)	Conclusion	How the gradient changes
\(\mathrm{f}'(a)=0\), \(\mathrm{f}''(a)<0\)	Local maximum	\(\mathrm{f}'\) is decreasing through zero
\(\mathrm{f}'(a)=0\), \(\mathrm{f}''(a)>0\)	Local minimum	\(\mathrm{f}'\) is increasing through zero

\(x\)	\(a^-\)	\(a\)	\(a^+\)
\(\mathrm{f}'(x)\)	Negative	0	Negative
\(\mathrm{f}''\)	Positive to zero to negative: concavity changes from up to down.

\(x\)	\(a^-\)	\(a\)	\(a^+\)
\(\mathrm{f}'(x)\)	Positive	0	Positive
\(\mathrm{f}''\)	Negative to zero to positive: concavity changes from down to up.

Second-Derivative Tests and Points of Inflection

Second-Derivative Tests and Points of Inflection

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