## Problem on constrained minimization

Suppose $x+ y+z=A$. What values of $x,y,z$ are such that $x^2 + y^2 + z^2$ is the smallest?
• ## Solution

We observe this is a constrained minimization problem: we are to minimize $x^2 + y^2 + z^2$ subject to the constraint $x+y+z=A$.
Recall that
First, we identify \begin{align}f(x,y,z) &= x^2 + y^2 + z^2 \\ g(x,y,z) &= x + y + z -A.\end{align}
Next, we write down the augmented functional: $$L(x,y,z,\lambda) = x^2 + y^2 + z^2 - \lambda (x + y + z - A).$$
From the first three equations, we conclude $x = \lambda /2$, $y = \lambda / 2$, and $z = \lambda / 2$. Hence, $x=y=z$. Now, from the bottom equation we get $$x=y=z=\frac{A}{3}.$$
The values of $x,y,z$ that minimize $x^2 + y^2 + z^2$ and satisfy $x+y+z=A$ are $x = y = z = A/3$.