To locate absolute extrema of a function on [a,b], which points should you examine?

To locate absolute extrema of a function on [a,b], which points should you examine?

• A. End Points and all critical points
• B. Endpoints only
• C. All interior points
• D. Points where f' is undefined only

Answer: (A)

To locate absolute extrema on a closed interval, you examine the endpoints and any interior points where the derivative is zero or undefined. This works because an absolute extremum on a closed interval can occur at the endpoints, or at interior points where the slope is flat or the function has a cusp/corner, which is where f′ is zero or does not exist. After listing those candidate points, evaluate the function at each one and compare the values to identify the absolute maximum and minimum. For example, with f(x) = -x^2 on [-1, 2], the derivative is -2x and vanishes at x = 0 (an interior critical point), giving f(0) = 0 as the absolute maximum, while f(-1) and f(2) are -1 and -4, respectively, so the absolute minimum is at x = 2. This shows why both endpoints and interior critical points must be checked.

To locate absolute extrema on a closed interval, you examine the endpoints and any interior points where the derivative is zero or undefined. This works because an absolute extremum on a closed interval can occur at the endpoints, or at interior points where the slope is flat or the function has a cusp/corner, which is where f′ is zero or does not exist. After listing those candidate points, evaluate the function at each one and compare the values to identify the absolute maximum and minimum. For example, with f(x) = -x^2 on [-1, 2], the derivative is -2x and vanishes at x = 0 (an interior critical point), giving f(0) = 0 as the absolute maximum, while f(-1) and f(2) are -1 and -4, respectively, so the absolute minimum is at x = 2. This shows why both endpoints and interior critical points must be checked.