Use Lagrange multipliers to find the minimum and maximum values of $y$ when $(x,y,z)$ is constrained to be in the intersection of the plane $x-y+2z=0$ and the ellipsoid $3x^2+2y^2+z^2=4$.
I am just confused that normally we have three functions and two of them are constraints. But in this one, there are just two and looks like they are both constraints. So how can I approach it? Thanks
You can use the equations $f_x=\lambda g_x+\mu h_x, f_y=\lambda g_y+\mu h_y, f_x=\lambda g_z+\mu h_z $ to get
$\;\;\;0=\lambda(1)+\mu(6x)$
$\;\;\;1=\lambda(-1)+\mu(4y)$
$\;\;\;0=\lambda(2)+\mu(2z)$.
Now use the 1st and 3rd equations to get a relation between $x$ and $z$, next solve for $y$, and substitute in the equation of the ellipsoid.
Alternatively, solve for $y$ to get $y=x+2z$, so substituting for $y$ in the equation of the ellipsoid gives the problem of finding the max and min of
$\;\;\;k(x,z)=x+2z$ subject to the constraint $4x^2+4xz+5z^2=4$.