Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = 6x4 − 8x3 − 24x2 + 1,    [−2, 3]absolute minimum value absolute maximum value

To find the absolute maximum and minimum values of a function on a given interval, we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 1: Find the derivative of the function f(x) = 6x^4 - 8x^3 - 24x^2 + 1.

f'(x) = 24x^3 - 24x^2 - 48x

Step 2: Set the derivative equal to zero and solve for x to find the critical points.

24x^3 - 24x^2 - 48x = 0

x(24x^2 - 24x - 48) = 0

x = 0, x = 2, x = -1

Step 3: Evaluate the function at the critical points and the endpoints of the interval [-2, 3].

f(-2) = 6(-2)^4 - 8(-2)^3 - 24(-2)^2 + 1 = 97 f(-1) = 6(-1)^4 - 8(-1)^3 - 24(-1)^2 + 1 = -25 f(0) = 6(0)^4 - 8(0)^3 - 24(0)^2 + 1 = 1 f(2) = 6(2)^4 - 8(2)^3 - 24(2)^2 + 1 = -31 f(3) = 6(3)^4 - 8(3)^3 - 24(3)^2 + 1 = 86

Step 4: Compare the function values to find the absolute maximum and minimum.

The absolute maximum value is 97 at x = -2. The absolute minimum value is -31 at x = 2.