`f(x) = x/(x - 5)` 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 numbers by setting the first derivative equal to zero and solving for the x value(x)
A critical number cannot be obtained using the first derivative. Critical values also exist where f(x) is not defined. Therefore there will be a critical number at x=5.
If f'(x)>0 the function is increasing in the interval.
If f'(x)<0 the function is decreasing in the interval.
Choose a value less than 5.
f'(4)=-5 Since f'(4)<0 the function is decreasing in the interval
Choose a value greater than 5.
f'(6)=-5 Since f'(6)<0 the function is decreasing in the interval (5, `oo).`
Because the direction of the function did not change, there are NO relative extrema.