Question

Is it possible for a continuous function to attain an absolute maximum on a closed interval if it has no critical numbers on the interval?

Answer 1

Yes it is possible for a continuous function to attain an absolute maximum on a closed interval,if it has no critical number because Extreme Value Theorem 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. It does not address how to find the extreme values.