记min{a，b，c}为a，b，c中最小值，若x，y是任意正实数，则M=min{x，1y，y+1x}的最大值为．

Question

记min{a，b，c}为a，b，c中最小值，若x，y是任意正实数，则M=min{x，(1)/(y)，y+(1)/(x)}的最大值为___．

Answer 1

设a=x，b=(1)/(y)，c=y+(1)/(x)=(1)/(a)+(1)/(b)．都大于0．
不妨设a≤b．则(1)/(a)≥(1)/(b)．
则(1)/(b)+(1)/(b)-b≤c-a=(1)/(a)+(1)/(b)-a≤(1)/(a)+(1)/(a)-a．
∴(2-b2)/(b)≤c-a≤(2-a2)/(a)．
①当a≥√(2)时，c≤a，此时c最小；
②当0√(2)