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

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).