John L. answered • 02/16/21
Naval Academy graduate with more than 10 years experience in teaching
The answer above, however, only points out where potential local max/mins exist. Taking the second derivative and evaluating at each point for concavity tells whether they are local max or local min. For example, the second derivative f"(x) = 12x +24. At x = 4, this is positive indicating that the curve is concave up at x = 4, making this a local min. Similarly, at x = -8, the second derivative is negative meaning it is concave down (or local max). Your question, however, asks not for the local max or min, but for the absolute. This is simply the highest or lowest value achieved on the specified interval. Because the curve is a positive cubic, the highest point must be where x = 5, and the lowest at x = -8. The highest value must then be f(5) and the lowest is f(-8). In other words, the local min is also the absolute min in this case.
