Question

Under what circumstances can you be certain that an absolute maximum and absolute minimum will exist for a function? (Hint: this has to do with limiting the function to a certain kind of set.) Give an example of a subset of R2 that fulfills the conditions, and an example of one that doesn’t. Describe how you find the maximum and minimum values of a function on such a set.

Answer 1

There is a well-known theorem for absolute maxima and minim, states that:

If f is continuous on a closed interval [a, b], then f has both an absolute maximum value and an absolute minimum value on the interval. This theorem says that a continuous function that is defined on a closed interval must have both an absolute maximum value and an absolute minimum value.