`f(x) = x^(2/3) - 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 value(s) by setting the first derivative equal to zero and solving for the x value(s).
Because 2=0 is not a true statement the first derivative will not produce a critical value.
If f'(x)>0 the function will increase in the interval.
If f'(x)<0 the function will decrease in the interval.
Notice f'(x)=undefined. This means the function is not differentiable at x=0.
Choose a value for x that is less than zero.
f'(-1)=-2/3 Since f'(-1)<0 the function is decreasing on the interval (-`oo,0).`
Choose a value for x that is greater than zero.
f'(1)=2/3 Since f'(1)>0 the function is increasing on the interval (0, `oo).`
Because the function changes direction from decreasing to increasing a relative minimum exists. The relative minimum exists at the point (0, -4) even though the derivative does not exist at that point.