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Thread: confusing lagrange multiplier question

  1. #1
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    confusing lagrange multiplier question

    f(x,y,z)=xy+z^3
    subject to x^2+y^2+z^2=1
    Find all the absolute maximum and minimum.


    I tried to solve the simutaneous equations, and I ended up with 14 critical points!
    So I doubt if I did right or wrong.
    Can anybody please help me and see how many of them? What are they? I solved that x=o, plus or minus square root 2 on 2, plus or minus 2/3
    y=0, plus or minus square root 2 on 2, plus or minus 2/3
    z=0, plus or minus 1, plus or minus 1/3
    Which rearrange them, I end up with 14 critical points. Am I right? Thanks a lot.
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  2. #2
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    The equations are

    $\displaystyle y=2 x \lambda$
    $\displaystyle x=2 y \lambda $
    $\displaystyle 3 z^2=2 z \lambda$
    $\displaystyle x^2+y^2+z^2=1$

    Right?
    If $\displaystyle x = 0$, so is $\displaystyle y = 0$. Then from the last equation $\displaystyle z = +1$ or $\displaystyle z = -1$. These are 2 points.
    If $\displaystyle z = 0$, then you get $\displaystyle x = \sqrt{2}/2$ or $\displaystyle x = -\sqrt{2}/2$ and $\displaystyle y = \sqrt{2}/2$ or $\displaystyle y = -\sqrt{2}/2$. All four combinations are valid, so you get another 4 points.

    The last option is if none of them are 0. Then \lambda is non-zero also because if it was, then we'd get one of $\displaystyle x,y,z = 0$. If you solve this you should get 4 solutions where $\displaystyle x = 2/3$, $\displaystyle y = 2/3$, $\displaystyle z = 1/3$ (where some of the signs change, i.e. $\displaystyle z = 1/3$, $\displaystyle x = 2/3$, $\displaystyle y = -2/3$)

    This is a grand total of 10 solutions. The way you did it is incorrect.
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