Answer
We show that the Lagrange equations have no solution.
By writing $f$ as a function of single variable subject to the constraint, we conclude that there is neither minimum nor maximum value of $f$.
Work Step by Step
We have $f\left( {x,y} \right) = x + y$ subject to the constraint $g\left( {x,y} \right) = x + 2y = 0$.
Using Theorem 1, the Lagrange condition $\nabla f = \lambda \nabla g$ yields
$\left( {1,1} \right) = \lambda \left( {1,2} \right)$
So, the Lagrange equations are
$1 = \lambda $, ${\ \ \ }$ $1 = 2\lambda $
We conclude that the Lagrange equations have no solution.
Using the constraint $x+2y=0$, we get $y = - \frac{x}{2}$. Substituting it in $f$ gives
$f\left( {x,y} \right) = \frac{x}{2}$
Thus, our problem may be stated alternatively by considering $f\left( {x,y} \right) = \frac{x}{2}$ subject to the constraint $g\left( {x,y} \right) = x + 2y = 0$. Since $x \in \mathbb{R}$, there is neither minimum nor maximum value of $f$.
