Find the absolute maximum value and the absolute minimum value,

g(x)=1/8 x^2 - 4 x^(1/2) on [0, 9]

g'(x) = x/4 - 2x^(-1/2) = x/4 - 2/√x

= 0 for a max/min

x/4 = 2/√x
x^(3/2) = 8
x = 8^(2/3) = 4
then f(4) = 2 - 4(2) = -6

for endpoints:
f(0) = 0
f(9) = 81/8 - 12 = -15/8 = -1.875

so within the given domain, the maximum function value is 0 , when x = 0

To find the absolute maximum and minimum values of the function g(x) = (1/8)x^2 - 4√x on the interval [0, 9], we need to follow these steps:

1. Take the derivative of the function g'(x) with respect to x.
- g'(x) = (1/4)x - 2/(√x)

2. Set the derivative equal to zero to find the critical points.
- (1/4)x - 2/(√x) = 0

3. Solve the equation to find the critical points.
- Multiply both sides of the equation by 4√x to eliminate the radical.
- x - 8 = 0
- x = 8

4. Check the endpoints of the interval [0, 9].
- Evaluate g(x) at x = 0 and x = 9.

Now, let's evaluate g(x) at the critical point and the endpoints:

- g(0) = (1/8)(0)^2 - 4√(0) = 0
- g(8) = (1/8)(8)^2 - 4√(8) = 8 - 4(2√2) = 8 - 8√2
- g(9) = (1/8)(9)^2 - 4√(9) = 81/8 - 12√1/8 = 81/8 - 3/2 = 75/8

Now, we compare these values to find the absolute maximum and minimum:

- The function has no maximum since g(8) = 8 - 8√2 is the largest value we found, but g(9) = 75/8 is even greater.
- The absolute maximum value of g(x) is g(9) = 75/8.

- The function has no minimum since g(0) = 0 is the smallest value we found, but g(8) = 8 - 8√2 is smaller.
- The absolute minimum value of g(x) is g(8) = 8 - 8√2.

Therefore, the absolute maximum value of g(x) is 75/8, and the absolute minimum value is 8 - 8√2.