对任意实数x,y,求S=x^2+2xy+3y^2+2x+6y+4的最小值
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对任意实数x,y,求S=x^2+2xy+3y^2+2x+6y+4的最小值
对任意实数x,y,求S=x^2+2xy+3y^2+2x+6y+4的最小值
对任意实数x,y,求S=x^2+2xy+3y^2+2x+6y+4的最小值
S=x^2+2xy+3y^2+2x+6y+4
S=(x+y)^2+2x+2y+1+2y^2+4y+3
S=(x+y)^2+2(x+y)+1+2(y+1)^2+1
S=(x+y+1)^2+2(y+1)^2+1>=1
当x+y+1=0
且y+1=0的时候,S取最小值1
即x=0,y=-1时,S最小为1
s=x^+2xy+3y^+2x+6y+4
=x^+2xy+y^+y^+6y+9+x^+2x+1-x^-6
=(x+y)^+y^+(y+3)^+(x+1)^-x^-6
s有最小值
x+y=0
y=0
y+3=0
x+1=0
x=0
x=0
y=0
s=4
x=-1
y=1
s=10
x=3
y=-3
s=10
当x=0,y=0时,s有最小值
s=4
S=x^2+2xy+3y^2+2x+6y+4
=x^2+2xy+y^2+2x+2y+1+2y^2+4y+2+1
=(x^2+2xy+y^2)+(2x+2y)+1+2(y^2+2y+1)+1
=(x+y)^2+2(x+y)+1+2(y+1)^2+1
=(x+y+1)^2+2(y+1)^2+1
因为对任意实数x,y,(x+y+1)^2≥0,2(y+1)...
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S=x^2+2xy+3y^2+2x+6y+4
=x^2+2xy+y^2+2x+2y+1+2y^2+4y+2+1
=(x^2+2xy+y^2)+(2x+2y)+1+2(y^2+2y+1)+1
=(x+y)^2+2(x+y)+1+2(y+1)^2+1
=(x+y+1)^2+2(y+1)^2+1
因为对任意实数x,y,(x+y+1)^2≥0,2(y+1)^2≥0
所以(x+y+1)^2+2(y+1)^2+1≥1
当且仅当x+y+1=0 且y+1=0
即x=0,y=-1时取等号,S取最小值1
所以S=x^2+2xy+3y^2+2x+6y+4的最小值为1
收起
x=0,y=-1时,S最小为1