The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin.
$f(x) = x^2 + y^2 + z^2$
$h(x) = x^2 + y^2 = 1$
$g(x) = x + z = 1$
The Attempt at a Solution
$\langle 2x, 2y, 2z \rangle = \lambda\langle2x, 2y,0\rangle + \mu\langle1,0,1\rangle$
This results in the equations:
$2x = 2x\lambda + \mu$
$2y = 2y\lambda$
$2z = \mu$
Then $\lambda = 1$, then $2x = 2x + \mu$, then $0 = \mu$, so then $z = 0$.
Then $x + z = 1$, so $x = 1$.
And then $1^2 + y^2 = 1$, so $y = 0$, so then I conclude that the point where this ellipse is furthest from the origin is $(1,0,0)$.
This is wrong. The answer should be $(-1,0,2)$