Question

Find the absolute extrema if they exist, as well as all values of x where they occur, for the function f(x) = (1/3)x³+(9/2)x²+18x+4 on the domain [-7,-2]. A. The absolute maximum is__ which occurs at x=

Answer 1

To find the absolute extrema of the function, the derivative is calculated and set to zero to find critical points. The function is then evaluated at these points and at the domain's endpoints to determine the absolute maximum and minimum values.

Step-by-step explanation:

To find the absolute extrema of the function f(x) = (1/3)x³ + (9/2)x² + 18x + 4 on the domain [-7,-2], we need to find the critical points of the function within the domain and evaluate the function at the endpoints of the domain. Here's how we proceed:

1. Find the derivative of the function: f'(x) = x² + 9x + 18.
2. Set the derivative equal to zero and solve for x to find critical points.
3. Evaluate the function f(x) at the critical points and at the endpoints of the given domain: x = -7 and x = -2.
4. Compare the values to determine which are the absolute maximum and minimum.

By examining the function at these points, we can find the absolute extrema, if they exist, and the values of x where they occur.