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A function f is continuous on the closed interval [-3,3] such that f(-3)=4 and f(3)=1. The function f'(x) and f"(x) have the properties given in the table below. (attached).

a. What are the x-coordinates of all absolute maximum and minimum points of f on the interval [-3,3]? Justify your answer.

b. What are the x-coordinates of all points on the inflection on the interval [-3,3]? Justify your answer.

Sorry for the delay in responding—I didn't see your update.

The slope is positive for x < -1 and negative or zero for x < -1, so the maximum occurs at (-1, f(-1)).

Where the slope is positive, f(x) ≥ 4.
The slope is negative or zero over (-1, 3], so the minimum occurs at (3, 1).
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There is only one point of inflection, when x = 1. The second derivative is positive when x < 1 and negative when x > 1, so there is a change in concavity at (1, f(1)).

There is no inflection at x = -1. (-1, f(-1)) is a cusp.

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