Question

1≤x^2+y^2≤,求证0.5≤x^2-xy+y^2≤3

Answer 1

题目漏了个2 假设x=asinb,y=acosb,其中1≤a≤2^0.5 x^-xy+y^ =a^2-a^2sinbcosb =a^2-1/2a^2sin2b 其中 -1≤sin2b≤1 那么 a^2-1/2a^2=1/2a^2≤a^2-1/2a^2sin2b≤a^2+1/2a^2=3/2a^2 则 1/2a^2≤x^-xy+y^≤3/2a^2 1/2*1≤x^-xy+y^≤3/2*2 则1/2≤x^-xy+y^≤3