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Math Help - Stumped on Maximum value Problem

  1. #1
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    Stumped on Maximum value Problem

    I want to find the maximum value of f(x)=x^a(1-x)^b on 0\leq x \leq 1 and a,b>0.

    I just apply the standard procedure for find the extrema of a function:

    f'(x)=ax^{a-1}(1-x)^b-b(1-x)^{b-1}

    ax^{a-1}(1-x)^b-b(1-x)^{b-1}=0

    ax^{a-1}(1-x)^b=b(1-x)^{b-1}

    I Divide both sides by (1-x)^b to obtain:

    ax^{a-1}=\frac{b}{1-x}

    This doesn't really seem to be going anywhere. I haven't been able to isolate x to obtain a function value. Any tips?
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  2. #2
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    Quote Originally Posted by adkinsjr View Post
    I want to find the maximum value of f(x)=x^a(1-x)^b on 0\leq x \leq 1 and a,b>0.

    I just apply the standard procedure for find the extrema of a function:

    f'(x)=ax^{a-1}(1-x)^b-b {\color{red} x^a} (1-x)^{b-1}

    ax^{a-1}(1-x)^b-b(1-x)^{b-1}=0

    ax^{a-1}(1-x)^b=b(1-x)^{b-1}

    I Divide both sides by (1-x)^b to obtain:

    ax^{a-1}=\frac{b}{1-x}

    This doesn't really seem to be going anywhere. I haven't been able to isolate x to obtain a function value. Any tips?
    I believe you've forgot a term (see above in red).
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  3. #3
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    Quote Originally Posted by Danny View Post
    I believe you've forgot a term (see above in red).
    thanks, I had a feeling this should be an easy one. I was able to isolate x after fixing the error.
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