`f(x) = x^4 - 32x + 4` 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(s).
The critical value is x=2.
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 for x that is less than 2.
f'(0)=-32 Since f'(0)<0 the function is decreasing on the interval (-`oo,2).`
Choose a value for x that is greater than 2.
f'(3)=76 Since f'(3)>0 the function is increasing on the interval (2, `oo).`
Since the function changed directions from decreasing to increasing a relative minimum exists. The relative minimum occurs at the point (2, -44).