设x、y、z都是不超过1的非负数，如果k=x+y(1-x)+z(1-x)(1-y)，试求k的取值范围．
【正确答案】：因为k=x+y(1-x)+z(1-x(1-y)所以1-k=1-x-y(1-x)-z(1-x)(1-y)=(1-x)[(1-y)-z(1-Y)]=(1-x)(1-y)(1-z)所以k=1-(1-x)(1-y)(1-x)因为0≤x≤10≤y≤10≤z≤1所以0≤1-x≤10≤1-y≤10≤1-z≤1所以≤(1-x)(1-y)(1-z)≤1-1≤-(1-x)(1-y)(1-z)≤0所以0≤1-(1-x)(1-y)(1-x)≤1即0≤k≤1．
