Question

Where can absolute maximum/minimum occur?

1) At critical points
2) At endpoints of the interval
3) At both critical points and endpoints
4) At neither critical points nor endpoints

Answer 1

Absolute maximums and minimums of a function can occur at critical points where the derivative is zero or undefined and at the endpoints of the interval. These points need to be evaluated to determine the global extrema.

So option (3) is the correct answer.

Step-by-step explanation:

Absolute maximums and minimums, also referred to as global extrema, of a function, can occur at critical points where the derivative is zero or does not exist, as well as at the endpoints of the interval over which the function is defined. It is important to check all these places to determine the global maximum and minimum values of a function.

Examples:

Critical points: Points on the graph of a function where the derivative is zero or undefined. These could indicate a local maximum or minimum, which may also be the absolute maximum or minimum if no other point on the function has a greater or smaller value.

Endpoints of the interval: If a function is defined on a closed interval, the absolute extrema could occur at the endpoints of that interval, since the endpoints are part of the function's domain.

Therefore, absolute maximums and minimums occur at both critical points and endpoints of the interval.