Question

What type of function families have maximum and minimum values? An exponential function does not have a maximum or minimum value why?

Answer 1

Explanation:

Any function that has a derivative that changes sign will have an extreme value (maximum or minimum). If the derivative never changes sign, the function will not have any extreme values.

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Logarithmic, exponential, and certain trigonometric, hyperbolic, and rational functions are monotonic, having a derivative that does not change sign. Odd-degree polynomials may also have this characteristic, though not necessarily. These functions will not have maximum or minimum values.

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Certain other trigonometric, hyperbolic, and rational functions, as well as any even-degree polynomial function will have extreme values (maximum or minimum). Some of those extremes may be local, and some may be global. In the case of trig functions, they may be periodic.

Composite functions involving ones with extreme values may also have extreme values.