Problem
ALG-B1-M07-P016 Same order
#16
★★★☆☆ Level 3 of 5
Let \(a\le b\le c\) and \(x\le y\le z\). Prove that \(ax+by+cz\ge az+by+cx\).
Cancel the common term \(by\) and look at the difference.
The difference is \(ax+cz-az-cx=(c-a)(z-x)\). Since \(c-a\ge0\) and \(z-x\ge0\), it is nonnegative. Therefore \(ax+by+cz\ge az+by+cx\).
A miniature version of the rearrangement inequality.