Question

chứng minh 5x^2 +2y^2 +4xy - 4x -y +5 >0 với mọi x

Answer 1

5x^2+2y^2+4xy-4x-y+5=(4x^2+y^2+4xy)+(x^2-4x+4)+(y^2-y+1/4)+3/4 =(2x+y)^2+(x-2)^2+(y-1/2)^2+3/4  (1)

 vi (2x+y)^2>=0 , (x-2)^2>=0  ,(y-1/2)^2>=0 (2)

tu 1 va 2 suy ra dieu phai chung minh

Answer 2

(x+y)^2+(x+2)^2-(-x-y)^2+x^2+y^2+1>=0