Find the absolute maximum and absolute minimum values of f on the given interval.
f(x)=(x^2-4)/(x^2+4), [-4,4]
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
To find the absolute maximum and absolute minimum values of a function on a given interval, we need to follow these steps:
1. Find the critical points within the interval (-4, 4) by taking the derivative of f(x) and setting it to zero.
2. Calculate the values of f(x) at the critical points and also at the endpoints of the interval.
3. Determine which of these values is the absolute maximum and which one is the absolute minimum.
Let's go through these steps one by one:
Step 1: Finding the critical points
To find the critical points, we first take the derivative of f(x) with respect to x:
f'(x) = [(2x)(x^2 + 4) - (x^2 - 4)(2x)] / (x^2 + 4)^2
After simplifying, we get:
f'(x) = (8x) / (x^2 + 4)^2
Setting f'(x) to zero:
8x = 0
x = 0
Step 2: Calculating the function values
Now we need to find the function values at the critical point (x = 0) and also at the interval endpoints (-4 and 4).
For the critical point, substituting x = 0 into f(x):
f(0) = (0^2 - 4) / (0^2 + 4) = -4 / 4 = -1
For the interval endpoints:
f(-4) = (-4^2 - 4) / (-4^2 + 4) = 12 / 16 = 3/4
f(4) = (4^2 - 4) / (4^2 + 4) = 12 / 20 = 3/5
Step 3: Determining the absolute maximum and minimum
To find the absolute maximum and minimum values, we compare the function values obtained:
- f(0) = -1 (critical point)
- f(-4) = 3/4 (endpoint)
- f(4) = 3/5 (endpoint)
The absolute maximum value is the largest value among the three, which is 3/4, obtained at x = -4. The absolute minimum value is the smallest value among the three, which is -1, obtained at x = 0.
Therefore, the absolute maximum value of f(x) on the interval [-4, 4] is 3/4, and the absolute minimum value is -1.