Find the absolute maximum and absolute minimum values of the function f(x)=x^3+6x^2-63x+4 on each of the indicated variables. Enter DNE for does not exist.

(A) Interval = [-8,0]
Absolute maximum =
Absolute minimum =
(B) Interval = [-5,4]
Absolute maximum =
Absolute minimum =
(C) Interval = [-8,4]
Absolute maximum =
Absolute minimum =

To find the absolute maximum and absolute minimum values of a function, we first need to find the critical points of the function within the given interval.

Step 1: Find the derivative of the function f(x).
Let's start with the given function f(x) = x^3 + 6x^2 - 63x + 4.
Taking the derivative will help us find the critical points.
f'(x) = 3x^2 + 12x - 63.

Step 2: Solve for the critical points.
To find the critical points, set the derivative f'(x) equal to zero and solve for x.
3x^2 + 12x - 63 = 0.

We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
(3x - 3)(x + 21) = 0.
Setting each factor equal to zero, we get x = 1 and x = -21.

Step 3: Check the endpoints of the interval.
The given interval is [-8, 0]. We need to evaluate the function at the endpoints to see if they yield the absolute maximum or minimum values.

f(-8) = (-8)^3 + 6(-8)^2 - 63(-8) + 4 = 512 + 384 + 504 + 4 = 1,404.
f(0) = 0^3 + 6(0)^2 - 63(0) + 4 = 4.

Step 4: Evaluate the function at the critical points.
We need to evaluate the function at the critical points x = 1 and x = -21.

f(1) = (1)^3 + 6(1)^2 - 63(1) + 4 = 1 + 6 - 63 + 4 = -52.
f(-21) = (-21)^3 + 6(-21)^2 - 63(-21) + 4 = -9,859 + 2,646 + 2,793 + 4 = -4,416.

Step 5: Identify the absolute maximum and minimum values.
Now, let's compare the values we have obtained:

(A) Interval = [-8,0]
Absolute maximum = 1,404 (at x = -8)
Absolute minimum = -52 (at x = 1)

(B) Interval = [-5,4]
Absolute maximum = To be determined
Absolute minimum = To be determined

(C) Interval = [-8,4]
Absolute maximum = To be determined
Absolute minimum = To be determined

To determine the absolute maximum and minimum values within the intervals in (B) and (C), we follow the same steps as above.