Question

If f(x) has an absolute maximum at x=4,an absolute minimum at x=5, a local maximum at x=2, and a local minimum at x=3,which of the following statements

A) The critical point(s) must be at x=2 and x=3.
B) The critical point(s) must be at x=4 and x=5.
C) There are no critical points for f(x).
D) The critical point(s) must be at x=3, x=4, and x=5.

Answer 1

The critical point(s) must be at x=2 and x=3. The correct answer isA) The critical point(s) must be at x=2 and x=3.

Step-by-step explanation:

To determine the critical points of the function, we need to find the values of x where the function has a local maximum or minimum. Since the function has a local maximum at x=2 and a local minimum at x=3, these are the critical points of the function.

If f(x) has an absolute maximum at x=4, an absolute minimum at x=5, a local maximum at x=2, and a local minimum at x=3, it means that the function changes direction at these points. By definition, a critical point is a point on a graph where the function's derivative is either zero or undefined, which typically corresponds to the peaks, bottoms, or flat regions in the function (where it changes direction).

However, the function also has an absolute maximum at x=4 and an absolute minimum at x=5. These points are not critical points because the function does not have a local maximum or minimum at these values. Therefore, the correct statement is:

The critical point(s) must be at x=2 and x=3. (Option A)