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It is given that $$f(x,y,z)=x^2-y^2+z^2$$ and the constraint is $$g(x,y,z)=x^2+2y^2+3z^2-1=0$$ I started to take Lagrange multiplier approach, here is my work $$ \left\{ \begin{array}{c} 2x-2xλ=0 \\ -2y-4yλ=0 \\ 2z-6zλ=0 \\ x^2+2y^2+3z^2-1=0 \end{array} \right. $$ Which gives $$ \left\{ \begin{array}{c} x(1-λ)=0 \\ y(1+2λ)=0 \\ z(1-3λ)=0 \\ x^2+2y^2+3z^2-1=0 \end{array} \right. $$ Then, I have no idea how to plug equation (1) to (3) in the constrain since x, y and z equal to zero. If x, y and z equal to zero, then equation (4) will not work. $$$$ If x, y and z not equal to zero, then $$ λ=1 \ or \ λ=-1/2 \ or \ λ=1/3 \ $$ What's wrong in my approach, do I missed other constraints?

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The system of equations that you obtained, that is $$ \begin{cases} x(1-\lambda)=0 \\ y(1+2\lambda)=0 \\ z(1-3\lambda)=0 \\ x^2+2y^2+3z^2-1=0 \end{cases} $$ is correct. Now looking at $x(1-\lambda)=0$, you should say that if $x\not=0$ then $\lambda=1$ and $$ \begin{cases} 3y=0 \\ -2z=0 \\ x^2+2y^2+3z^2-1=0 \end{cases} \implies y=z=0,\; x=\pm 1.$$ Can you take it from here and see what happens when $y\not=0$? What if $z\not=0$?

Note that the constraint $x^2+2y^2+3z^2-1=0$ implies that at least one of $x$, $y$, $z$ is different from zero.

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so far you calculation is correct

now for each value of $\lambda$ you have to find a solution

then you will check the value of f(x,y,z) for each one and determine max/min)

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