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 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.

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