The plane $x + y + 2z = 30$ intersects the paraboloid $z = x^2 + y^2$ in an ellipse.
Find the points on the ellipse that are nearest to and farthest from the origin.
I know you have to find the gradient for $f$, but after that, I have no clue how to proceed. Thank you in advance.
Best Answer
let $d^2(x,y,z)=x^2+y^2+z^2$.
you want $\nabla (d^2)$ to be in the plane of $\nabla f$ and $\nabla g$. That is, for some $\lambda$ and $\mu$, you want $$\nabla(d^2)+\lambda\nabla f+\mu\nabla g=0.$$