Question

Find all open intervals on which (x^2)/(x^2+4) is decreasing.

Answer 1

Step 1: Take the derivative
Step 2: Solve for x when the derivative = 0
Step 3: Interval test to see when the derivative is negative or the derivative is positive. When the derivative is negative, that means that the function is decreasing.

Answer 2

Let me know if you get stuck.

Answer 3

Steve, I think it would be better to just set f'(x)<0, for step 2.

Answer 4

(of course we all agree a derivative is necessary) ... NY NY, have you found the derivative?

Answer 5

I got x=0
And to do the interval test, do i plug in a number less than and greater than 0?

Answer 6

What was the derivative of the function?

Answer 7

I got (8x)/((x+4)^2)

Answer 8

You mean (x^2+4)^2 ?

Answer 9

@SolomonZelman I guess you could set f'(x) < 0, but to find the values, I think you are essentially doing the same thing by interval tests.

Answer 10

Oh yes, i wrote it wrong.

Answer 11

Yes ... I'm just giving a suggestion, because as soon as you set f'(x)<0, it becomes clear.

Answer 12

The function is decreasing when the slope (or, the derivative) is negative, right?

Answer 13

For this reason, to see where the function is decreasing, we set \(f'(x)<0\).
Or, in your case,
\(\color{black}{\displaystyle \frac{8x}{(x^2+4)^2} <0 }\)

Answer 14

Well, we know \(\color{black}{\displaystyle (x^2+4)^2>0 }\) (for all \(\color{black}{\displaystyle x }\)), so in terms of "being grater than zero, all what's going to matter is whether or not \(\color{black}{\displaystyle 8x }\) is negative or not. In other words, whether or not \(\color{black}{\displaystyle 8x<0 }\).

Answer 15

8x is less than 0 when x is less than 0(?)

Answer 16

YEs!

Answer 17

Ok.
So
F(x) is decreasing when x<0
?

Answer 18

Or (-infinity,0)

Answer 19

Yes!

Answer 20

Thanks