Second derivative test

Statement

What this test is for

This test is a partial test (i.e., it may be inconclusive) for determining whether a given critical point for a function is a point of local minimum, point of local maximum, or neither.

What the test states

Suppose $f$ is a function and $c$ is a point in the interior of the domain of $f$, i.e., $f$ is defined on some open interval containing $c$. Suppose, further, that $f''$, i.e., the second derivative of $f$, exists at $c$. Suppose also that $f'(c)=0$, so $c$ is a critical point for $f$. Then:

Hypothesis	Conclusion
$f''(c)<0$	$f$ attains a local maximum value at $c$ (the value is $f(c)$)
$f''(c)>0$	$f$ attains a local minimum value at $c$ (the value is $f(c)$)
$f''(c)=0$	The test is inconclusive. $f$ may attain a local maximum value, a local minimum value, have a point of inflection, or have some different behavior at the point $c$.

Related tests

• First derivative test
• Higher derivative tests