Find Min: K= x ² + 2y ² +2xy + 2x- 2y + 2015 su dung da thuc :(a+b+c) ² October 25, 2020 by Sigridomena Find Min: K= x ² + 2y ² +2xy + 2x- 2y + 2015 su dung da thuc :(a+b+c) ²
$K=x^2+2y^2+2xy+2x-2y+2015$ $=(x^2+y^2+1+2xy+2x+2y)+(y^2-4y+4)+2010$ $=(x+y+1)^2+(y-2)^2+2010$ Vì $(x+y+1)^2≥0∀x,y$ $(y-2)^2≥0∀y$ $⇒(x+y+1)^2+(y-2)^2+2010≥2010∀x,y$ Dấu ”=” xảy ra khi $\left \{ {{x+y+1=0} \atop {y-2=0}} \right.⇔\left \{ {{x=-3} \atop {y=2}} \right.$ Vậy $K_{min}=2010⇔\left \{ {{x=-3} \atop {y=2}} \right.$. Reply
$K=x^2+2y^2+2xy+2x-2y+2015$
$=(x^2+y^2+1+2xy+2x+2y)+(y^2-4y+4)+2010$
$=(x+y+1)^2+(y-2)^2+2010$
Vì $(x+y+1)^2≥0∀x,y$
$(y-2)^2≥0∀y$
$⇒(x+y+1)^2+(y-2)^2+2010≥2010∀x,y$
Dấu ”=” xảy ra khi $\left \{ {{x+y+1=0} \atop {y-2=0}} \right.⇔\left \{ {{x=-3} \atop {y=2}} \right.$
Vậy $K_{min}=2010⇔\left \{ {{x=-3} \atop {y=2}} \right.$.