Use Lagrange multipliers to find extremes of $2x+y$ constrained to an ellipsoid

Question

use Lagrange multipliers to find the maximum and minimum values of $$ 2x+y $$ on the ellipsoid $$ x^2/a^2+y^2/b^2+z^2/c^2=1 $$

My issue is creating two functions that are suitable to use in a Lagrange multiplier process ie the first seems to be a function of two variables while the second seems to be a function of three.

I am also unsure if what is meant by $2x+y$ is $2x+y=0$ or $2x+y=z$.

If anyone could clarify I would greatly appreciate it.

Answer 1

In this context, because the ellipsoid is a function of three variables, I would interpret the objective function also as a function of three variables: $$ F(x, y, z) = 2x + y $$ Stated formally, the problem becomes:

Find the extremal values of the objective $F(x, y, z) = 2x + y$ subject to the constraint $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$.