Question

一道数学不等式题目

已知x,y∈R，满足（x²+y²+2)(x²+y²-1)-18≤0.求证：xy≤2

Answer 1

（x²+y²+2)(x²+y²-1)≤18等价于（x²+y²)^2+(x²+y²)≤20
2xy*(2xy+1)≤（x²+y²)^2+(x²+y²)≤20
(2xy+1/2)^2-1/4≤20
(2xy+1/2)^2≤81/4
2xy+1/2≤9/2
2xy≤4
xy≤2

Answer 2

（x²+y²+2)(x²+y²-1)≤18等价于（x²+y²)^2+(x²+y²)≤20
2xy*(2xy+1)≤（x²+y²)^2+(x²+y²)≤20
(2xy+1/2)^2-1/4≤20
(2xy+1/2)^2≤81/4
2xy+1/2≤9/2
2xy≤4
xy≤2
清楚吧