# Difference between revisions of "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 says

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 $c$.

### Relation with critical points

The second derivative test is specifically used only to determine whether a critical point where the derivative is zero is a point of local maximum or local minimum. Note in particular that:

• For the other type of critical point, namely that where $f'$ is undefined, the second derivative test cannot be used.
• Since point of local extremum implies critical point, we don't have to worry about points that are not critical points -- none of them will give local extrema.

## Strength of the test

### Second derivative test requires twice differentiability at but not around the point

The second derivative test can be applied at a critical point $c$ for a function $f$ only if $f$ is twice differentiable at $c$. This in particular forces $f$ to be once differentiable around $c$.

However, the test does not require $f''$ to be defined around $c$ or to be continuous at $c$.