Math

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.

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  1. 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

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    posted by Steve

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