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Math Help - maximum value of f(x) for each positive x

  1. #1
    rcs
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    maximum value of f(x) for each positive x

    Can anybody guide me on this problem?

    thank you.
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  2. #2
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    Re: maximum value of f(x) for each positive x

    Let y = x + 1/x. Consider y^3 and express x^3 + 1/x^3 through y. Then express x^6 + 1/x^6 through (x^3 + 1/x^3)^2 and thus through y.
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    Re: maximum value of f(x) for each positive x

    you mean to say that i am going to substitute the numerator first quantity by y and so it is y^3 then the second quantity would be solely y?
    then the denominator as well? pls guide me on this sir so confused to find what does maximum value
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    Re: maximum value of f(x) for each positive x

    Quote Originally Posted by rcs View Post
    you mean to say that i am going to substitute the numerator first quantity by y and so it is y^3 then the second quantity would be solely y?
    then the denominator as well? pls guide me on this sir so confused to find what does maximum value
    Substitute y = x + \frac{1}{x} just like emakarov suggested. So (x + \frac{1}{x})^3 = y^3.


    Also, (x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}). Substitute x + \frac{1}{x} to obtain


    y^3 = x^3 + \frac{1}{x^3} + 3y \Rightarrow x^3 + \frac{1}{x^3} = y^3 - 3y


    Then you can substitute this into the original f(x).
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    Re: maximum value of f(x) for each positive x

    thank you i got what it meant... but how about the denominator sir? do i need to substitute it also? do i need only to manipulate the x^6 + 1/x^6?
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    Re: maximum value of f(x) for each positive x

    Yeah, you're gonna have some more algebraic manipulation. Hint: What is (x^3 + \frac{1}{x^3})^2?
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    Re: maximum value of f(x) for each positive x

    little help pls still cant get the point
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    Re: maximum value of f(x) for each positive x

    Quote Originally Posted by richard1234 View Post
    Yeah, you're gonna have some more algebraic manipulation. Hint: What is (x^3 + \frac{1}{x^3})^2?
    i got it x^6 +1/x^6 + 2.... need little guide here sir... may i know the next step ?
    thank u
    Last edited by rcs; July 30th 2012 at 06:47 PM.
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    Re: maximum value of f(x) for each positive x

    Quote Originally Posted by rcs View Post
    i got it x^6 +1/x^6 + 2.... need little guide here sir... may i know the next step ?
    thank u
    Substitute into the denominator..
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    Re: maximum value of f(x) for each positive x

    i made the hint given by richard1234 (x^3 + 1/x^3) ^2

    and so i have obtained this

    f(x)= (y^3 + y^3 - 3y) / (y^6 - (y^3 - 3y)^2 - 2 - 2)

    = (y^3 + y^3 - 3y) / (y^6 - ( y^6 + 6y^4 + 9y^2) - 4
    = (y^3 + y^3 - 3y) / 6 y^4 - 9y^2 - 4
    = y(2y^2 - 3) / 3y^2 ( 2y^2 - 3) - 4

    im stuck here ...

    need your help
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    Re: maximum value of f(x) for each positive x

    Eh, that looks right, as long as your algebra's correct.

    Now you have a function in terms of y. Maximize it.
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    Re: maximum value of f(x) for each positive x

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