Minimum Norm Under a Linear Constraint

Use a Lagrange multiplier with the constraint $x+y=1$.

By symmetry, the optimal point should have $x=y$.

1. 1. Set up the Lagrangian

   Introducing a multiplier for the constraint, the first-order conditions give $2x=\mu$ and $2y=\mu$, so $x=y$.

2. 2. Solve using the constraint

   Combining $x=y$ with $x+y=1$ gives $x=y=1/2$, and the minimum value follows by substitution.

   $$x=y=\frac{1}{2},x^2+y^2=\frac{1}{2}$$

Answer: x=1/2, y=1/2, minimum value 1/2