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Usually at the minimum or maximum of a function, one of the following conditions arises:

  • The derivative is zero.
  • The derivative is undefined.
  • The point is at the end-points of the domain that is being considered (or of the naturally-defined domain, for example, zero for the square root).

This will give you "candidate points"; to find out whether each of these candidate points actually is a maximum or a minimum, additional analysis is required. For example, if the second derivative is positive, you have a minimum, if the second derivative is negative, you have a maximum - but if it is zero, it may be a maximum, a minimum, or neither.

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Q: How do you know if a point is a maximum or a minimum?
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