Second derivative test: Difference between revisions
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Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the interior of the domain of <math>f</math>, i.e., <math>f</math> is defined on some [[open interval]] containing <math>c</math>. Suppose, further, that <math>f''</math>, i.e., the [[second derivative]] of <math>f</math>, exists at <math>c</math>. Then: | Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the interior of the domain of <math>f</math>, i.e., <math>f</math> is defined on some [[open interval]] containing <math>c</math>. Suppose, further, that <math>f''</math>, i.e., the [[second derivative]] of <math>f</math>, exists at <math>c</math>. Suppose also that <math>f'(c)=0</math>, so <math>c</math> is a [[critical point]] for <math>f</math>. Then: | ||
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Revision as of 13:34, 2 May 2012
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 is a function and is a point in the interior of the domain of , i.e., is defined on some open interval containing . Suppose, further, that , i.e., the second derivative of , exists at . Suppose also that , so is a critical point for . Then:
| Hypothesis | Conclusion |
|---|---|
| attains a local maximum value at (the value is ) | |
| attains a local minimum value at (the value is ) | |
| The test is inconclusive. may attain a local maximum value, a local minimum value, have a point of inflection, or have some different behavior at the point . |
