The derivative is not defined at x = ±4. Setting the derivative equal to zero, we get
2(16 − x) = 2x^232 − 2x^2 = 2x^232 = 4x^2x = ±If you’re wondering why we don’t use the negative root, it’s
because there is no such thing as a negative area.Note that the domain of this function is −4 ≤ x ≤ 4, so these numbers serve as endpoints of the interval.
Let’s compare the critical values and the endpoints.
When x = −4, y = 0 and the area is 0.When x = 4, y = 0 and the area is 0.When x = , y = and the area is 16.Thus, the maximum area occurs when x = and the area equals 16.
Try some of these solved problems on your own. As always, cover the answers as you work.
PROBLEM 1. A rectangular field, bounded on one side by a building, is to be fenced in on the other three
sides. If 3,000 feet of fence is to be used, find the dimensions of the largest field that can be fenced in.
Answer: First, let’s make a rough sketch of the situation.