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First derivative theorem for local extreme values

Autor:Steve Phelps
This interactive figure illustrates the first derivative theorem for local extreme values:
  • If has a local maximum or minimum value at an interior point of its domain, and if is defined at , then .
The local extrema at input values x = c of the function's domain are labeled A, B, C, D, and E (if they all exist). The New Function button will change the graph of the function.  Since the derivative of this function is defined at each local maximum or local minimum point shown, we can conclude that at each of these local extrema.
Developed for use with Thomas' Calculus, published by Pearson.

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