find the critical numbers and the open intervals on which the function is increasing or decreasing

f(x)= x/((x)^2+4)

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Let us take the derivative of the given function using Quotient Rule.

Let set: u = x. So, du = 1

v = x^2 + 4, dv = 2x

Use the formula for the Quotient Rule.

`(1*(x^2+4)-x(2x))/((x^2+4)^2) = (x^2 + 4 - 2x^2)/((x^2+4)^2)`

Combine like terms.

`(4 - x^2)/((x^2+4))`

Equate the top to zero to solve for the critical numbers.

4 - x^2 = 0.

Add x^2 on both sides.

4 = x^2

Take the square root of both sides.

x = 2, -2.

So, we will have intervals: (-inf, -2), (-2, 2), and (2, inf).

Let us plug-in x = -3, 0, 3 on the f '(x).

f' (-3) = (4 - (-3)^2)/((-3)^2 + 4)^2) = -5/169

f' (0) = (4 - (0)^2)/((0)^2 + 4)^2) = 1/4

f' (3) = (4 - (3)^2)/((3)^2 + 4)^2) = -5/169

Hence, **it is increasing in the interval (-2, 2), and decreasing on intervals **

**(-inf, -2) and (2, inf)**.

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