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Math Help - optimization with constraint

  1. #1
    Senior Member
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    Jun 2009
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    251

    optimization with constraint

    question: z=100x^{.25}y^{.75} where x=48-4y Use chain rule to find \frac{dz}{dy}. Solve \frac{dz}{dy}=0 for y in terms of x and use the constraint to find x and y to maximize z.
    NOTE: these are normal derivatives, not partial

    work:
    z=100(48-4y)^{.25}y^{.75}
    I take the derivative and subsitute x back in to get \frac{dz}{dy}=-x^{-.75}y^{.75}+\frac{3}{4}y^{-.25}x^{.25}
    Then I set it equal to 0 to get \frac{3}{4}y^{-.25}x^{.25}=x^{-.75}y^{.75}
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  2. #2
    MHF Contributor
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    Looks good so far. To solve for y, use power properties. y^a*y^b=y^{a+b}. You can multiply both sides by y^a, where a will both cancel one y term and make the other y^1, which is very convenient.
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