Problem
ALG-B2-M01-P002 Fixed sum
#2
★★☆☆☆ Level 2 of 5
If \(x+y=14\), find the smallest possible value of \(x^2+y^2\).
Hint 1. Express it through \(x+y\) and \(x-y\).
Hint 2. Use \((x-y)^2\ge0\).
\[x^2+y^2=\frac{(x+y)^2+(x-y)^2}{2}=98+\frac{(x-y)^2}{2}\ge98.\] The minimum is \(98\), attained at \(x=y=7\).
Module training problem. Method tags: squares, equality-case.