Difference between revisions of "Second derivative test"

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===What the test states===
 
===What the test states===
  
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:
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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>. 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 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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