Let g(x)=(4x)/(x^2+1) on the interval [-4,0]. Find the absolute maximum and absolute minimum of g(x) on this interval.

The absolute max occurs at x=.
The absolute min occurs at x=

Hey I did a couple. Now you do some.

WOW

To find the absolute maximum and minimum of a function on a given interval, you need to follow these steps:

1. Find the critical points of the function, which are the points where the derivative is either zero or undefined.
2. Evaluate the function at the critical points and at the endpoints of the interval.
3. Compare the values obtained in step 2 to find the absolute maximum and minimum.

Now let's find the absolute maximum and minimum of g(x)=(4x)/(x^2+1) on the interval [-4,0].

Step 1: Find the critical points
To find the critical points, we need to find where the derivative of g(x) is either zero or undefined. Let's first find the derivative of g(x):

g'(x) = [(4)(x^2+1) - (4x)(2x)] / (x^2+1)^2
= [4x^2 + 4 - 8x^2] / (x^2+1)^2
= (-4x^2 + 4) / (x^2+1)^2

To find the critical points, we need to solve g'(x) = 0. However, notice that the numerator will never equal zero since -4x^2 + 4 is never equal to zero. Therefore, g'(x) will only be undefined at the points where the denominator (x^2+1)^2 is equal to zero.

(x^2+1)^2 = 0
x^2 + 1 = 0

This equation has no real solutions, so there are no critical points where the derivative is undefined.

Step 2: Evaluate the function at critical points and endpoints
Since there are no critical points, we only need to evaluate the function at the endpoints of the interval [-4,0].

g(-4) = (4*(-4))/((-4)^2 + 1) = -16/17
g(0) = (4*0)/(0^2 + 1) = 0/1 = 0

Step 3: Compare the values obtained
Now, we compare the values we obtained in Step 2.

-16/17 is the absolute minimum of g(x) on the interval [-4,0], which occurs at x = -4.
0 is the absolute maximum of g(x) on the interval [-4,0], which occurs at x = 0.

So, the absolute maximum occurs at x = 0, and the absolute minimum occurs at x = -4.