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Math Help - Find minimum value for function to be invertible?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find minimum value for function to be invertible?

    Fin the minimum value of a and b for which f(x) = x^x ; f : [a,\infty] \rightarrow [b,\infty) to be an invertible function.
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  2. #2
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    A function will be invertible as long as it does not have a "turning point"- as long as its derivative is not 0.
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    Super Member fardeen_gen's Avatar
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    I know that for a fact. What do we do in the given question?
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    MHF Contributor Bruno J.'s Avatar
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    f(x) = e^{x\log x}

    f'(x) = e^{x\log x}(1/x+1)

    now solve for f'(x)=0
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  5. #5
    Senior Member pankaj's Avatar
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    Quote Originally Posted by Bruno J. View Post
    f(x) = e^{x\log x}

    f'(x) = e^{x\log x}(1/x+1)

    now solve for f'(x)=0
    f'(x)=x^x(1+\ln x)

    f'(x)=0

    x=\frac{1}{e}

    f\left(\frac{1}{e}\right)=\frac{1}{e^{\frac{1}{e}}  }

    Therefore, for f to be invertible, f:\left[\frac{1}{e},\infty\right)\rightarrow\left[\frac{1}{e^{\frac{1}{e}}},\infty\right)

    Thus a=\frac{1}{e} and b=\frac{1}{e^{\frac{1}{e}}}
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