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Math Help - Differentiation Problem

  1. #1
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    Differentiation Problem

    Let a>b>0 and let n\in\mathbb{N} satisfy n\geq 2 . Prove that a^{1/n} - b^{1/n} < (a-b)^{1/n}

    I am given the following hint:
    Show that f(x) := x^{1/n} - (x-1)^{1/n} is decreasing for x\geq 1 , and evaluate f at 1 and a/b
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  2. #2
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    Quote Originally Posted by Kipster1203 View Post
    Let a>b>0 and let n\in\mathbb{N} satisfy n\geq 2 . Prove that a^{1/n} - b^{1/n} < (a-b)^{1/n}

    I am given the following hint:
    Show that f(x) := x^{1/n} - (x-1)^{1/n} is decreasing for x\geq 1 , and evaluate f at 1 and a/b
    It suffices to prove this for a fixed but arbitrary a,b>0

    b^{\frac{1}{n}}-a^{\frac{1}{n}}<(b-a)^{\frac{1}{n}}\Leftrightarrow \frac{\left(b^{\frac{1}{n}}-a^{\frac{1}{n}}\right)^n}{b-a}=\frac{\left(\left(\frac{b}{a}\right)^{\frac{1}{  n}}-1\right)^n}{\frac{b}{a}-1}<1.

    But, the LHS is clearly \frac{f\left(\frac{b}{a}\right)-f(1)}{\frac{b}{a}-1} where f(x)=\left(x^{\frac{1}{n}}-1\right)^n. Thus, by the mean value we have that the LHS equals f'(c)=\frac{c^{\frac{1}{n}}\left(c^{\frac{1}{n}}-1\right)^{n-1}}{c}=c^{\frac{n-1}{-n}}\left(c^{\frac{1}{n}}-1\right)^{n-1}=\left(1-c^{\frac{-1}{n}}\right)^{n-1} for some c\in\left[1,\tfrac{b}{a}\right]. But, clearly then 0\leqslant 1-c^{\frac{-1}{n}}<1\implies 0<\left(1-c^{\frac{-1}{n}}\right)^{n-1}<1.

    The conclusion follows.
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  3. #3
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    Quote Originally Posted by Kipster1203 View Post
    Let a>b>0 and let n\in\mathbb{N} satisfy n\geq 2 . Prove that a^{1/n} - b^{1/n} < (a-b)^{1/n}

    I am given the following hint:
    Show that f(x) := x^{1/n} - (x-1)^{1/n} is decreasing for x\geq 1 , and evaluate f at 1 and a/b


    Show that f'(x)<0\,\,\forall x>1 and then 1<a\slash b\Longrightarrow f(1)<f(a\slash b) and you're done (caution : beware of negative powers!)

    Tonio
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