An inflection point for a function f is a point on the graph of f where the concavity changes. First, let us find the intervals where f is concave up (or concave down.) For this, we use the sign of the second derivative. We have
f´(x) = 3x2 - 24x, and
f ´´(x) = 6x - 24.
The function will be concave up when f´´(x) > 0, so we solve the resulting inequality:
6x - 24 > 0
6x > 24
x > 4.
Thus, the graph of f(x) is concave up on the interval (4,∞).
Similarly, f is concave down when f´´(x) < 0, so solving 6x - 24 < 0 gives x < 4. Therefore, f is concave down on the interval (-∞,4). Since the function changes from concave up to concave down at x = 4, the function has an inflection point at x = 4. (Choice 2.)
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