Question

What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f?

Answer 1

In general, there is no guarantee of the existence of an absolute maximum or minimum value of `f`.

Explanation:

If `f` is continuous on a closed interval `[a,b]` (that is: on a closed and bounded interval), then the Extreme Value Theorem guarantees the existence of an absolute maximum or minimum value of `f` on the interval `[a,b]`.