Question

Find the absolute maximum and absolute minimum values of f on the given interval.

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

Answer 1

use the quotient rule,

f ' (x) = ( 2x(x^2 + 4) - 2x(x^2 - 4))/(x^2 + 4)^2
= (2x^3 + 8x - 2x^3 + 8x)/(x^2 + 4)^2
= 16x/(x^2+4)^2
= 0
---> x = 0

if x = 0 , f(0) = -4/4 = -1
(0,-1) is a local min by looking at the graph

http://www.wolframalpha.com/input/?i=f%28x%29%3D%28x%5E2-4%29%2F%28x%5E2%2B4%29

but since there is a domain, we have to look at the end points of the domain
by summetry,
f(±4) = (16-4)/(16+4)
= 12/20
= 3/5

so (-4,3/5) and (4,3/5) are maximum points

since it asked for the min and max value of f

the min is -1, and the max is 3/5