HMMT 二月 2004 · CALC 赛 · 第 8 题

HMMT February 2004 — CALC Round — Problem 8

If x and y are real numbers with ( x + y ) = x − y , what is the maximum possible value of y ?

解析

If x and y are real numbers with ( x + y ) = x − y , what is the maximum possible value of y ? √ 3 Solution: 3 2 / 16 2 ◦ By drawing the graph of the curve (as shown), which is just a 135 clockwise rotation 4 and scaling of y = x , we see that the maximum is achieved at the unique point where 3 dy/dx = 0. Implicit differentiation gives 4( dy/dx + 1)( x + y ) = 1 − dy/dx , so setting √ √ √ 3 3 3 3 4 dy/dx = 0 gives 4( x + y ) = 1. So x + y = 1 / 4 = 2 / 2, and x − y = ( x + y ) = 2 / 8. √ √ √ 3 3 3 Subtracting and dividing by 2 gives y = ( 2 / 2 − 2 / 8) / 2 = 3 2 / 16 . 0.5 1 1.5 -0.5 -1 -1.5