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Math Help - Decreasing function

  1. #1
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    Decreasing function

    f(x)=x^\frac{1}{n} - (x-1)^\frac{1}{n}

    I need help showing this is decreasing for x> 1

    Any suggestions? trying to do this with the derivative.

    I get

    f'(x)=\frac{x^\frac{1-n}{n}-(x-1)^\frac{1-n}{n}}{n}

    Can I assume x^\frac{1-n}{n}<(x-1)^\frac{1-n}{n}, If so why?

    Any help would greatly be appreciated.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by hockey777 View Post
    f(x)=x^\frac{1}{n} - (x-1)^\frac{1}{n}

    I need help showing this is decreasing for x> 1

    Any suggestions? trying to do this with the derivative.

    I get

    f'(x)=\frac{x^\frac{1-n}{n}-(x-1)^\frac{1-n}{n}}{n}

    Can I assume x^\frac{1-n}{n}<(x-1)^\frac{1-n}{n}, If so why?

    Any help would greatly be appreciated.
    Ok so we need to show that x^{\frac{1}{n}}-(x-1)^{\frac{1}{n}}\geq(x+1)^{\frac{1}{n}}-(x)^{\frac{1}{n}}\forall{x}\in[1,\infty)

    Now we may rewrite this as

    x^{\frac{1}{n}}\geq\frac{(x-1)^{\frac{1}{n}}+(x-1)^{\frac{1}{n}}}{2}

    Or

    \left(\frac{(x+1)+(x-1)}{2}\right)^{\frac{1}{n}}\geq\frac{(x+1)^{\frac{  1}{n}}+(x-1)^{\frac{1}{n}}}{2}

    and since

    \frac{d^2}{dx^2}\bigg[x^{\frac{1}{n}}\bigg]=\frac{x^{\frac{1}{n}}}{x^2}\bigg[\frac{1}{n^2}-\frac{1}{n}\bigg]

    Which is less than zero for all x and n greater than one...therefore x^{\frac{1}{n}} is concave \forall(x,n)\in\mathbb{R^+}

    Therefore f\left(\frac{x+y}{2}\right)\geq\frac{f(x)+f(y)}{2}  \quad\blacksquare
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  3. #3
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    I should be more clear that I'm trying to prove that

    a^\frac{1}{n}-b^\frac{1}{n}<(a-b)^\frac{1}{n}

    For a>b>0 and n>2[/tex]

    by showing that function is decreasing, I can prove this. Therefore I can't use a few of your assumptions in example.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by hockey777 View Post
    I should be more clear that I'm trying to prove that

    a^\frac{1}{n}-b^\frac{1}{n}<(a-b)^\frac{1}{n}

    For a>b>0 and n>2[/tex]

    by showing that function is decreasing, I can prove this. Therefore I can't use a few of your assumptions in example.
    I did show that your function is decreasing
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  5. #5
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    Isn't what you did show that

    f(x+1)<f(x)

    not that f(x_2)< f(x_1)
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