To find the absolute maximum and absolute minimum values of the function $f(x) = x^3 - 6x^2 - 63x + 5$ over each of the given intervals, we need to follow these steps:

1. Find the critical points of $f(x)$ by setting its derivative $f'(x)$ equal to zero.
2. Evaluate $f(x)$ at the critical points and the endpoints of the given intervals.
3. Compare these values to determine the absolute maximum and minimum values over each interval.

Step 1: Find the critical points

First, we find the derivative of $f(x)$: $f'(x) = 3x^2 - 12x - 63$

Set the derivative equal to zero to find the critical points: $3x^2 - 12x - 63 = 0$

Divide the equation by 3: $x^2 - 4x - 21 = 0$

Solve the quadratic equation: $x = \frac{4 \pm \sqrt{16 + 84}}{2}$ $x = \frac{4 \pm \sqrt{100}}{2}$ $x = \frac{4 \pm 10}{2}$

So, the critical points are: $x = 7$ $x = -3$

Step 2: Evaluate $f(x)$ at the critical points and endpoints of each interval

(a) Interval = $[-4, 0]$

Endpoints: $x = -4$ and $x = 0$ Critical points within the interval: $x = -3$

Calculate $f(x)$ at these points: $f(-4) = (-4)^3 - 6(-4)^2 - 63(-4) + 5 = -64 - 96 + 252 + 5 = 97$ $f(0) = 0^3 - 6(0)^2 - 63(0) + 5 = 5$ $f(-3) = (-3)^3 - 6(-3)^2 - 63(-3) + 5 = -27 - 54 + 189 + 5 = 113$

1. Absolute maximum = 113 (at $x = -3$) 2. Absolute minimum = 5 (at $x = 0$)

(b) Interval = $[-1, 8]$

Endpoints: $x = -1$ and $x = 8$ Critical points within the interval: $x = 7$

Calculate $f(x)$ at these points: $f(-1) = (-1)^3 - 6(-1)^2 - 63(-1) + 5 = -1 - 6 + 63 + 5 = 61$ $f(8) = 8^3 - 6(8)^2 - 63(8) + 5 = 512 - 384 - 504 + 5 = -371$ $f(7) = 7^3 - 6(7)^2 - 63(7) + 5 = 343 - 294 - 441 + 5 = -387$

1. Absolute maximum = 61 (at $x = -1$) 2. Absolute minimum = -387 (at $x = 7$)

(c) Interval = $[-4, 8]$

Endpoints: $x = -4$ and $x = 8$ Critical points within the interval: $x = -3$ and $x = 7$

Calculate $f(x)$ at these points: $f(-4) = (-4)^3 - 6(-4)^2 - 63(-4) + 5 = 97$ $f(8) = 8^3 - 6(8)^2 - 63(8) + 5 = -371$ $f(-3) = (-3)^3 - 6(-3)^2 - 63(-3) + 5 = 113$ $f(7) = 7^3 - 6(7)^2 - 63(7) + 5 = -387$

1. Absolute maximum = 113 (at $x = -3$) 2. Absolute minimum = -387 (at $x = 7$)

Summary:

(a) Interval = $[-4, 0]$ 1. Absolute maximum = 113 (at $x = -3$) 2. Absolute minimum = 5 (at $x = 0$)

(b) Interval = $[-1, 8]$ 1. Absolute maximum = 61 (at $x = -1$) 2. Absolute minimum = -387 (at $x = 7$)

(c) Interval = $[-4, 8]$ 1. Absolute maximum = 113 (at $x = -3$) 2. Absolute minimum = -387 (at $x = 7$)

Would you like more details or have any questions?

Suggested Questions:

1. How do you find the derivative of a function?
2. What are critical points and why are they important?
3. How do you solve a quadratic equation?
4. Can you explain how to evaluate a function at a given point?
5. What is the difference between an absolute maximum and a relative maximum?

Tip:

When finding the maximum and minimum values of a function on a closed interval, always check both the critical points within the interval and the values at the endpoints.