In calculus, the second derivative test is a criterion often useful for determining whether a given critical point of a function is a local maximum or a local minimum using the value of the second derivative at the point.
The test states: If the function f is twice differentiable at a critical point x, meaning that, then:
- If then is concave down at .
- If then is concave up at .
- If, shows no concavity or is a possible inflection point.
In the last case, although the function may have a local maximum or minimum at x, because the function is sufficiently "flat" (i.e. ) the extremum is rendered undetected by the second derivative. In this case one has to examine the third derivative. The point at which is an inflection point if concavity changes on either side of it. For example, (0,0) is an inflection point on because, and and .
Read more about Second Derivative Test: Multivariable Case, Proof of The Second Derivative Test, Concavity Test
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