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Second derivative test for extreme values

Author:Marc Renault
Suppose is a point in the domain of where both and . This means that is a critical point of . Furthermore,
  1. If and at , then has a local maximum at .
  2. If and at , then has a local minimum at .
  3. If at , then has a saddle point at .
  4. If at , then the test is inconclusive.
In the interactive figure, you can enter a function , then rotate the view to see where its extreme values lie. Several examples are given, or you can define your own function. Drag the red point to set , and consider the relationship between the derivatives of at and the shape of the -graph.
Developed for use with Thomas' Calculus and Interactive Calculus, published by Pearson.

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