`f(x) = (x - 1)^2(x+ 3)` 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 values by setting the first derivative equal to zero and solving for the x value(s).
The critical numbers are x=-5/3 and x=1.
If f'(x)>0 the function increases in the interval.
If f'(x)<0 the function decreases in the interval.
Choose an x value less than -5/3.
f'(-2)=3 Since f'(-2)>0 the function increases in the interval (-`oo,-5/3).`
Choose an x value between -5/3 and 1.
f'(-1)=-4 Since f'(-1)<0 the function decreases in the interval (-5/3, 1).
Choose an x value greater than 1.
f'(2)=11 Since f'(2)>0 The function increases in the interval (1, `oo).`
Because the direction of the function changed from increasing to decreasing a relative maximum will exist at x=-5/3. The relative maximum is the point
Because the direction of the function changed from decreasing to increasing a relative minimum will exist at x=1. The relative minimum is the point (1, 0).