Final answer:
To find the location of the absolute maximum and absolute minimum of a function on a given interval, find the critical points and endpoints of the function within that interval, evaluate the function at those points, and determine the maximum and minimum values.
Step-by-step explanation:
To find the location of the absolute maximum and absolute minimum of a function on a given interval, we need to find the critical points and endpoints of the function within that interval. First, find the derivative of the function and set it equal to zero to find the critical points. Then, evaluate the function at the critical points and the endpoints to determine the absolute maximum and minimum.
For example, if the function is f(x) = x^2 on the interval [0,4], we find the derivative f'(x) = 2x. Setting this equal to zero, we get 2x = 0, so x = 0. Evaluating the function at x = 0, we get f(0) = 0^2 = 0. Next, we evaluate the function at the endpoints of the interval, which are f(0) = 0^2 = 0 and f(4) = 4^2 = 16. Since f(4) = 16 is greater than both f(0) = 0 and f(0) = 0, 4 is the location of the absolute maximum.
Therefore, the location of the absolute maximum is x = 4 and the location of the absolute minimum is x = 0.