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Let x and y be real numbers such that \(x^2 - xy + y^2 = 3. \)
Let m be the minimum value of \(x^2 + y^2,\) and let M be the maximum value of \(x^2 + y^2.\)Find m+M

 Feb 24, 2019
 #1
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\(\text{There are a few ways to go about this. Lagrange multipliers is one}\\ f = x^2 + y^2\\ g = (x^2 - xy+y^2 - 3)\\ \text{Solve }\nabla(f+\lambda g)=0\)

 

\(\text{This results in 3 equations}\\ 2x + \lambda(2x-y)=0\\ 2y+\lambda(2y-x) = 0\\ x^2 - xy +y^2 = 3\)

 

\(\text{This system has 4 solutions}\\ (x,y) = \pm(1,-1)\\ (x,y) = \pm (\sqrt{3},\sqrt{3})\\ \text{resulting in min and max of }x^2 + y^2 \text{ being }2 \text{ and } 6\\ m+M=2+6 = 8\)

 

I leave the gritty details to you.

 Feb 25, 2019

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