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Thread: Find the maximum and minimum values of

  1. #1
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    Find the maximum and minimum values of

    Find the maximum and minimum values of on the disk D: .
    maximum value:
    minimum value:
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  2. #2
    MHF Contributor
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    Well, you can probably see by inspection that (x,y)=(0,0) is a minimum, but let's look at the critical points - where $\displaystyle \frac{\partial{f}}{\partial{x}}=\frac{\partial{f}} {\partial{y}}=0$.

    $\displaystyle \frac{\partial{f}}{\partial{x}}=12x$

    $\displaystyle \frac{\partial{f}}{\partial{y}}=14y$

    so the only critical point is (0,0). But we have to look at the boundary, that is, where $\displaystyle x^2+y^2=1$. Solving for $\displaystyle y^2$ and substituting gives:

    $\displaystyle 6x^2+7y^2=6x^2+7(1-x^2)=7-x^2$, which has a maximum at x=0, and minima at x=1 and -1. The minima are greater than the minimum at (0,0), so we can ignore them. But the maximum at x=0 gives us maxima at (x,y)=(0,1) and (0,-1).

    - Hollywood
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