Question

Interval of increase x^(1/3)(x+4)

Answer 1

Your function f(x)=(x^(1/3))(x+4)

What your going to want to do first is calculate your first derivative.
You will want to use both the power rule and the product rule.

f '(x)=(x+4)/x^(2/3)+x^(1/3)

f '(x)=4/3(x^(-2/3))(x+1)

Now find the critical numbers

4/3(x^(-2/3))=0 x+1=0
                 x=0, x=-1

Now plug numbers on the intervals (-infinity, -1),(-1,0) and (0,infinity) into the derivative of your function. The numbers I would choose are -2, -1/2 and 1.

Since f '(-2) is negative, f '(-1/2) is positive and f '(1) is positive, (-1,-3) and (1,5) are relative extrema. This means that the function f(x)=(x^(1/3))(x+4) is increasing on the interval (-1,0) U (0,infinity).