Question

Consider the function f(x)=6x3−54x2+144x f ( x ) = 6 x 3 − 54 x

2 + 144 x on the interval [−2,7] [ − 2 , 7 ] . Find the absolute extrema for the function on the given interval.

Answer 1

To find the absolute extrema of the function f(x)=6x^3-54x^2+144x on the interval [-2,7], we need to find the maximum and minimum values of the function within that interval.

Step-by-step explanation:

To find the absolute extrema of the function f(x)=6x^3-54x^2+144x on the interval [-2,7], we need to find the maximum and minimum values of the function within that interval.

1. First, we find the critical points of the function by taking the derivative and setting it equal to zero: f'(x) = 18x^2 - 54x + 144 = 0.
2. Next, we evaluate the function at the critical points and the endpoints of the interval: f(-2), f(7), and the solutions to f'(x) = 0.
3. The largest value among these points is the maximum, and the smallest value is the minimum of the function on the interval.

By finding the critical points and evaluating the function at these points and the endpoints, we can determine the absolute extrema of the function on the given interval.