HMMT 二月 2004 · CALC 赛 · 第 8 题

英文解析

8. If x and y are real numbers with ( x + y ) = x − y , what is the maximum possible valueof y ?
   √
   Solution: 3 2 / 163
   °2
   By drawing the graph of the curve (as shown), which is just a 135 clockwise rotationand scaling of y = x , we see that the maximum is achieved at the unique point where 4
   dy/dx = 0. Implicit differentiation gives 4( dy/dx + 1)( x + y ) = 1 − dy/dx , so setting 3
   √ √ √
   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