题目

首届八年级数学学科能力解题技能展示决赛试题!
设x,y,z都是不超过1的非负实数,若k=x+y(1-x)+z(1-x)(1-y),试求k的取值范围.

提问时间：2020-10-30

答案

K=X+Y-XY+Z(1-X-Y+XY)
=(X+Y-XY)+Z-Z(X+Y-XY)
=(1-Z)(X+Y-XY)+Z-1+1
=(1-Z)(X+Y-XY-1)+1
=(1-Z)((X-1)-Y(X-1))+1
=(1-Z)(1-Y)(X-1)+1
由题目已知
0≤1-Z≤1
0≤1-Y≤1
-1≤X-1≤0
所以-1≤K-1≤0
0≤K≤1