Explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = x^3 - 6x^2 - 63x + 2 over the indicated intervals, we need to evaluate the function at the critical points and endpoints of each interval. Let's calculate each value step by step:
(a) Interval = [-4, 0]
1. To find the critical points, we take the derivative of the function and set it equal to zero:
f'(x) = 3x^2 - 12x - 63
Setting f'(x) = 0 and solving for x:
3x^2 - 12x - 63 = 0
Solving this quadratic equation, we find two critical points: x = -3 and x = 7.
2. Next, we evaluate the function at the critical points and endpoints of the interval:
f(-4) = (-4)^3 - 6(-4)^2 - 63(-4) + 2 = -102
f(0) = 0^3 - 6(0)^2 - 63(0) + 2 = 2
f(-3) = (-3)^3 - 6(-3)^2 - 63(-3) + 2 = -20
f(7) = (7)^3 - 6(7)^2 - 63(7) + 2 = -532
The absolute maximum value is the highest value among these: 2
The absolute minimum value is the lowest value among these: -532
Therefore, for the interval [-4, 0]:
Absolute maximum = 2
Absolute minimum = -532
(b) Interval = [-1, 8]
1. We already found the critical points in the previous step, which are x = -3 and x = 7.
2. Now, we evaluate the function at the critical points and endpoints of the interval:
f(-1) = (-1)^3 - 6(-1)^2 - 63(-1) + 2 = 70
f(8) = (8)^3 - 6(8)^2 - 63(8) + 2 = -254
The absolute maximum value is the highest value among these: 70
The absolute minimum value is the lowest value among these: -254
Therefore, for the interval [-1, 8]:
Absolute maximum = 70
Absolute minimum = -254
(c) Interval = [-4, 8]
1. Again, the critical points are x = -3 and x = 7.
2. Evaluating the function at the critical points and endpoints:
f(-4) = (-4)^3 - 6(-4)^2 - 63(-4) + 2 = -486
f(8) = (8)^3 - 6(8)^2 - 63(8) + 2 = -254
The absolute maximum value is the highest value among these: -254
The absolute minimum value is the lowest value among these: -486
Therefore, for the interval [-4, 8]:
Absolute maximum = -254
Absolute minimum = -486