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

$$f(x)=$$$$\frac{x}{x^2+9}$$

I think I needed to use the quotient rule so i applied it and got up to :

$$\frac{-x^2+9}{(x^2+9)^2}$$

if this is right, what do I need to do next?

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You've correctly used the quotient rule to find the derivative or gradient of $f(x)$.

The question then asks for two things: critical points and open intervals on which the function is either increasing or decreasing.

For any point $x$ in the domain of the function, one of three things is true. Either:

  1. $f'(x) > 0$, i.e. the function is increasing;
  2. $f'(x) = 0$, i.e. the function is a critical point;
  3. $f'(x) < 0$, i.e. the function is decreasing.

You can find the points which fall into category 2; any other points will fall into open intervals, each of which will either satisfy category 1, increasing, or category 3, decreasing.

If you take your domain, the reals, and remove the critical points, you'll be left with just open intervals. Each of these has $f(x)$ either increasing on all of it or decreasing on all of it. This is because your derivative is a continuous function, meaning that it can't jump from being negative (category 3) to being positive (category 1) without being 0 at some point in between.

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You're about halfway there. Now determine when the derivative is zero; these are the critical points. Since the denominator is always positive (for real $x$), this amounts to determining when the numerator is zero. Finally, plug in some values of $x$ between the critical points to identify where the derivative is positive or negative, and hence, where the curve is increasing or decreasing.

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