Calculus of a Single Variable, Chapter 3, 3.3, Section 3.3, Problem 27

Given: f(x)=x^(1/3)+1
Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s).
f'(x)=(1/3)x^(-2/3)=0
(1)/(3x^(2/3))=0
1=0
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.