Second derivative test

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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.

Related tests

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.