`f(x) = x^2/(x^2 - 9)` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.
Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s).
A critical value is at 0. Critical values also exist where f(x) is not defined. Therefore there are also critical values at 3 and -3.
If f'(x)>0 the function is increasing on the interval.
If f'(x)<0 the function is decreasing on the interval.
Choose a value for x that is less than -3.
Choose a value for x that is between -3 and 0.
Choose a value for x that is between 0 and 3.
Choose a value for x that is greater than 3.
The function increases in the interval (-`oo,-3).`
The function increases in the interval (-3, 0).
The function decreases in the interval (0, 3).
The function decreases in the interval (3, `oo).`
Because the function changes direction from increasing to decreasing, there is a relative maximum at x=0. The relative maximum occurs at the point (0, 0).