Final answer:
To find the extreme values of the function f(x) = x^4 - 72x^2 + 10 on a given interval, one needs to find the critical points, evaluate the function at these points and at the interval's endpoints, and compare these values to get the absolute minimum.
Step-by-step explanation:
To find the extreme values of the function f(x) = x^4 - 72x^2 + 10 on the interval [-5, 13], you have to find the critical points of the function and evaluate f at the endpoints of the interval and at these critical points.
Step 1: Find the derivative of the function f'(x) = 4x^3 - 144x. Set the derivative equal to zero and solve for x to find critical points.
Step 2: Solve f'(x) = 0 to get x = 0 and x = ± 6. However, only x = 0 is within the interval [-5, 13].
Step 3: Compute f(x) at the critical point and endpoints:
f(-5), f(0), f(13).
Step 4: The smallest value among f(-5), f(0), and f(13) gives the absolute minimum value. If a smaller value does not exist within the interval, then the absolute minimum value does not exist (DNE).
After computing, you compare these values and select the smallest one as the absolute minimum value.