Use the given function and the given interval below to answer parts a through c.

Question

Use the given function and the given interval below to answer parts a through c. f(x) = -x^2 + 11 on [-3, 4] a. Find the critical point(s) of f on the given interval b. Determine the absolute extreme values of f on the given interval when they exist. c. Use a graphing utility to confirm your conclusions. a. Select the correct choice below and, if necessary fill in the answer box to complete your choice. A. The critical point(s) is (are) at x = B. There are no critical points of f on the given interval b. What is/are the absolute maximum/maxima of f on the given interval? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The absolute maximum/maxima is/are at x = B. There is no absolute maximum of f on the given interval. What is/are the absolute minimum/minima of f on the given interval? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The absolute minimum/minima is/are at x = B. There is no absolute minimum of f on the given interval. c. Use a graphs utility to confirm your conclusions. Determine which graph represents f(x). Choose the correct answer below.

Explanation / Answer

a)
f(x) = -x^2+11
f'(x) = -2x
f'(x) = 0 at x=0
So x=0 is a criticalpoint
Other critical points are end points
Answer:
The criticalpoints are x=-3,0,4

b)
f'(x) is 0 at x=0
f''(x) = -2
so, x=0 is maxima
f(0)=-0^2+11 = 11

minima occurs at end points
f(-3) = - (-3)^2+11 = 2
f(4) = - 4^2+11 = -5

The absolute maximum is 11 at x = 0
The absolute minimum is -5 at x=4

c)
The correct graph is graph D
Answer: D

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