Question

Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 2x3 − 6x2 − 18x + 2, [−2, 4] Step 1 The absolute maximum and minimum values of f occur either at a critical point inside the interval or at an endpoint of the interval. Recall that a critical point is a point where f '(x) = 0 or is undefined. We begin by finding the derivative of f

Answer 1

Given

f(x) = 2x³ - 6x² - 18x + 2

its derivative is

f'(x) = 6x² - 12x - 18

Then f has critical points when

6x² - 12x - 18 = 6 (x² - 2x - 3) = 6 (x - 3) (x + 1) = 0

or when x = -1 and x = 3. Because f is a polynomial, it and its derivatives are defined everywhere.

Classify the critical points by checking the sign of the second derivative at each one:

f''(x) = 12x - 12

• At x = -1, we have f''(-1) = -24 < 0, which indicates a local maximum at the point (-1, f(-1)) = (-1, 12).

• At x = 3, we have f''(3) = 24 > 0, which indicates a local minimum at (3, f(3)) = (3, -52).

We also check the value of f at the endpoints of the given domain.

• At x = -2, the graph of f passes through the point (-2, f(-2)) = (-2, -2).

• At x = 4, f goes through the point (4, f(4)) = (4, -38).

So, over the interval [-2, 4], we have

• an absolute maximum of 12 when x = -1, and

• an absolute minimum of -52 when x = 3

Answer 2

x=-0.2

Explanation: