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Math Help - Determine the minimizers and maximizers of f subject to the given constraints

  1. #1
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    Determine the minimizers and maximizers of f subject to the given constraints

    f(x,y) = 3*(x^3) + 2*(y^3) subject to x^2 + y^2 = 4. This should be done using Lagrange multipliers, but i couldn't solve it.
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    \nabla f(x,y)=9x^2\mathbf{i}+6y^2\mathbf{j}

    \lambda\nabla g(x,y)=2x\lambda\mathbf{i}+2y\lambda\mathbf{j}

    \displaystyle 9x^2=2x\lambda\Rightarrow \frac{9x}{2}=\lambda

    \displaystyle 6y^2=2y\lambda\Rightarrow 6y^2=2y\left(\frac{9x}{2}\right)

    \displaystyle 6y=9x\Rightarrow x=\frac{2}{3}y

    \displaystyle x^2+y^2=4\Rightarrow \left(\frac{2}{3}y\right)^2+y^2=4

    \displaystyle \frac{4y^2}{9}+y^2=\frac{13y^2}{9}=4

    \displaystyle 13y^2=36\Rightarrow y=\pm\frac{6}{\sqrt{13}}

    \displaystyle x=\left(\frac{1}{3}\right)\left(\frac{\pm 6}{\sqrt{13}}\right)=\pm\frac{2}{\sqrt{13}}

    \displaystyle f\left(\frac{2}{\sqrt{13}},\frac{6}{\sqrt{13}}\rig  ht)=\cdots

    \displaystyle f\left(\frac{-2}{\sqrt{13}},\frac{-6}{\sqrt{13}}\right)=\cdots
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  3. #3
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    Quote Originally Posted by enrique View Post
    f(x,y) = 3*(x^3) + 2*(y^3) subject to x^2 + y^2 = 4. This should be done using Lagrange multipliers, but i couldn't solve it.
    Could you please show what you tried? I see nothing at all unusual about this problem.

    For any "Lagrange multiplier" problem in two variables, you will get two equations of the form g(x,y)= \lambda f(x,y) and h(x,y)= \lambda k(x,y).

    Dividing one equation by the other gives an equation that does not involve \lambda: \frac{g(x,y)}{h(x,y)}= \frac{f(x,y)}{k(x,y)} which, in this case, is particularly simple.
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  4. #4
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    This is one form of Lagrange multipliers, that is taught in calculus courses. This question is from the optimization book of Nash & Sofer, they use some different notation and solve similar types of questions using matrices, which is confusing for me. But i see that this type of problems can be solved by just using Calculus "Lagrange multipliers", right?
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