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Math Help - Decreasing

  1. #1
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    Decreasing

    Let a>b>0 and let n be in N (natural numbers) satisfy n greater than or equal to 2. Prove that:
    a^(1/n)-b^(1/n) < (a-b)^(1/n) by showing that x^(1/n)-(x-1)^(1/n) is decreasing for all x greater than or equal to 1 and using that property appropriately.

    Thank you for any help.
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  2. #2
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    Taking the derivative of what they gave x^(1/n)-(x-1)^(1/n),
    I got:

    (1/n)[(x^((1-n)/n))-(x-1)^((1-n)/n)). How do I show that this is decreasing?

    I'm assuming that then I can say that using this,
    a=x
    b=x-1 (since b is less than a)
    and use the derivatives accordingly. Also how can I conclude that the right side is less than the left in the original equation I want to prove? Is it just because it's decreasing?
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  3. #3
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    Quote Originally Posted by dave007rules View Post
    How do I show that this is decreasing?
    Since the function is differenciable you show that the derivative is zero.
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