2 bài toán khá cân não

K

koumancu

M

microwavest

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$3x^2-2xy-2x+11y^2+6y-1$

$=3x^2-2x(y+1)+11y^2+6y-1$

$=3[x^2-\dfrac{2}{3}x(y+1)]+11y^2+6y-1$

$=3[x^2-2.x.\dfrac{1}{3}(y+1)+\dfrac{1}{9}(y+1)^2-\dfrac{1}{9}(y+1)^2]+11y^2+6y-1$

$=3[x-\dfrac{1}{3}(y+1)]^2-\dfrac{1}{3}(y+1)^2+11y^2+6y-1$

$=3[x-\dfrac{1}{3}(y+1)]^2+\dfrac{32}{3}y^2+\dfrac{16}{3}y-\dfrac{4}{3}$

$=3[x-\dfrac{1}{3}(y+1)]^2+\dfrac{32}{3}(y^2+2.y.\dfrac{1}{4}+\dfrac{1}{16}-\dfrac{1}{16})-\dfrac{4}{3}$

$=3[x-\dfrac{1}{3}(y+1)]^2+\dfrac{32}{3}(y+\dfrac{1}{4})^2-2$ \geq $-2$

Vậy Min=$-2$ \Leftrightarrow $x=\dfrac{1}{4}$ và $y=-\dfrac{1}{4}$
 
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