find the absolute maximum and minimum values of f on the given interval

f(x)= (x^2-4)/(x^2+4), [-4, 4]

find f'(x)

set f'(x)=0 to find the critical values.
when looking for the min and max include the intervals with the critical numbers

f'(x)= [2x(x^2+4)-(2x(x^2-4)]/[(x^2+4)^2]
simplify....
and set equal to zero

To find the absolute maximum and minimum values of the function f(x) = (x^2-4)/(x^2+4) on the interval [-4, 4], we can follow these steps:

1. Identify the critical points by finding where the derivative of f(x) is zero or undefined.
2. Determine the endpoints of the interval [-4, 4].
3. Evaluate the function f(x) at these critical points and the endpoints.
4. Compare the values obtained in step 3 to find the absolute maximum and minimum values.

Let's start by finding the critical points of f(x):

1. Find the derivative of f(x):
f'(x) = (2x(x^2+4) - 2(x^2-4)(2x))/(x^2+4)^2
= (2x(x^2+4) - 4x(x^2-4))/(x^2+4)^2
= (8x)/(x^2+4)^2

2. Set f'(x) equal to zero and solve for x:
(8x)/(x^2+4)^2 = 0
8x = 0
x = 0

So, x = 0 is a potential critical point. However, we also need to check for any values where the derivative is undefined.

3. Determine if f'(x) is undefined:
The derivative f'(x) is defined for all values of x, so there are no additional critical points.

Now, let's evaluate the function f(x) at these critical points and the endpoints of the interval:

Evaluate f(x) at x = -4:
f(-4) = ((-4)^2 - 4) / ((-4)^2 + 4)
= (16 - 4) / (16 + 4)
= 12 / 20
= 3/5

Evaluate f(x) at x = 0:
f(0) = (0^2 - 4) / (0^2 + 4)
= (-4) / 4
= -1

Evaluate f(x) at x = 4:
f(4) = ((4)^2 - 4) / ((4)^2 + 4)
= (16 - 4) / (16 + 4)
= 12 / 20
= 3/5

Now, we compare the values obtained:

f(-4) = 3/5
f(0) = -1
f(4) = 3/5

The absolute maximum value is 3/5, and it occurs when x = -4 and x = 4 on the interval [-4, 4]. The absolute minimum value is -1, which occurs at x = 0.

Therefore, the absolute maximum of f(x) on the interval [-4, 4] is 3/5, and the absolute minimum is -1.