Question

实数x、y满足x^2+xy+y^2=1,则F(x，y)=x^3y+xy^3的最大值与最小值之和为？

Answer 1

x^3y+xy^3
=xy（x^2+y^2）
=xy(1-xy)
=-(xy-1/2)^2+1/4
又xy=1-（x^2+y^2）≤1-2xy
∴xy≤1/3
x^2+xy+y^2=1=(x+y)^2-xy≥-xy
∴xy≥-1当x=1，y=-1或y=1，x=-1可取到等号
∴最大值与最小值之和为
-（-1-1/2）^2+1/4-（1/3-1/2）^2+1/4=-16/9

Answer 2

G(x,y)=x^3y+xy^3+a(x^2+xy+y^2-1)
Gx`(x,y)=3x^2y+y^3+a(2x+y)=0
Gy`(x,y)=x^3+3xY^2+a(x+2y)=0
3x^2y+y^3+a(2x+y)-{x^3+3xY^2+a(x+2y)}=0
(x-y)(-x^2-y^2+2xy+a)=0
1)x=y
x^2+xy+y^2=1
3x^2=1
x=±1/√3
y=±1/√3
F(x，y)=x^3y+xy^3=2/9