[toán 9]tim min

L

lp_qt

A=x2+y2+xy−5x−4y+2002A=x^2+y^2+xy-5x-4y+2002

\Leftrightarrow 2A=2x2+2y2+2xy−10x−8y+40042A=2x^2+2y^2+2xy-10x-8y+4004

=(x2+y2+9+2xy−6x−6y)+(x2−4x+4)+(y2−2y+1)+3990=(x^2+y^2+9+2xy-6x-6y)+(x^2-4x+4)+(y^2-2y+1)+3990

=(x+y−3)2+(x−2)2+(y−1)2+3990≥3990=(x+y-3)^2+(x-2)^2+(y-1)^2+3990 \ge 3990

dấu = xảy ra \Leftrightarrow x=2;y=1x=2;y=1

\Rightarrow A≥1995A\ge 1995
 
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H

hien_vuthithanh

A=x2+y2+xy−5x−4y+2002A=x^2 +y^2 +xy -5x -4y+2002

\Leftrightarrow x2+(y−5)x+y2−4y+2002−A=0x^2+(y-5)x+y^2-4y+2002-A=0

PT trên có nghiệm \Leftrightarrow Δ=(y−5)2−4y2+16y−8008+4A≥0\Delta=(y-5)^2-4y^2+16y-8008+4A \ge 0

\Leftrightarrow 4A≥3(y−1)2+7980≥26604A\ge 3(y-1)^2+7980 \ge 2660

\Rightarrow A≥1995A\ge 1995

\Rightarrow Min tại x=2;y=1x=2 ;y=1
 
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