Minimizing Sum of Squares Under a Linear Constraint

Use a Lagrange multiplier with the constraint $x+y=2$.

By symmetry of the objective and constraint, the optimum should have $x=y$.

1. 1. Set up the Lagrangian conditions

   The first-order conditions give $2x=\mu$ and $2y=\mu$, so $x=y$.

2. 2. Apply the constraint

   Combining $x=y$ with $x+y=2$ gives $x=y=1$, and substituting into the objective gives the minimum.

   $$x=y=1,x^2+y^2=2$$

Answer: x=y=1, minimum value 2