`f(x) = x^(1/3) + 1` 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).
1=0 is not a true statement. A critical value cannot be found using the first derivative.
If f'(x)>0 the function increases on the interval.
If f'(x)<0 the function decreases on the interval.
The domain for the function is all real values for x.
Notice that f'(0)=undefined. This means the slope of the function at x=0 does not exist.
Choose an x value less than 0.
f'(-1)=1/3 Since f'(-1)>0 the graph is increasing on the interval (-`oo,0).`
Choose an x value greater than 0.
f'(1)=1/3 Since f'(1)>0 the graph is increasing on the interval (0, `oo).`
Since the function does not change direction there will not be a relative extrema.