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How do I determine the maximum and minimum values (and points) of a given function along a curve?

Ex: $f(x,y)=xy$; $5x^2+5y^2+6xy - 64=0$

Using Lagrange multipliers, I got to this system of equations, but I don't see any easy way to solve it:

$$y = \lambda (10x+6y)$$ $$x = \lambda (10y+6x)$$ $$5x^2 + 5y^2 + 6xy = 64$$

Thank you!

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  • $\begingroup$ Lagrange multipliers is the tool $\endgroup$ – Claude Leibovici Oct 31 '15 at 12:47
  • $\begingroup$ That's what I tried, but I get to a really complicated system of equations :/... $\endgroup$ – Vinícius Lopes Simões Oct 31 '15 at 19:06
  • $\begingroup$ It is simple. Please, post your work in order one can see where and/or why you are stuck and be able to help you. $\endgroup$ – Claude Leibovici Oct 31 '15 at 19:16
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You're on the right track!

With the equations you have, you just have to solve for lambda.

So:

\begin{equation} \lambda = \frac{y}{10x+6y} \end{equation} and \begin{equation} \lambda = \frac{x}{10y+6x} \end{equation}

Equate them to each other, and simplify. You'll see that the relationship between them is \begin{equation} \label{eq} x^{2} = y^{2} \end{equation} Solve for either x or y, your choice, and plug that into the constraint equation so that you can solve for one variable. Then, relate back to ($x^{2}=y^{2}$) to find the other variable. You will get some points, which you just plug into your $f$ to find the maximum/minimum values.

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