f(x)=x−9x^(1/3)
I need help finding where function increases and decreases. I found the critical numbers which are -3sqrt(3), 3sqrt(3), and 0.
f' = 1 - 3/x^(2/3)
= (x^(2/3)-3)/x^(2/3)
f'=0 when
x^(2/3) = 3
x^2 = 27
x = ±√27
the tangent is vertical at x=0.
f increases where f' > 0: (-∞,√27)U(√27,∞)
f decreases where f' < 0: (-√27,√27)
The graph below confirms this:
https://www.wolframalpha.com/input/?i=x%E2%88%929x^%281%2F3%29