Toán Lớp 8: tìm gtnn x² – xy + y² – 2x – 2y
ko chép mạng nha . cảm ơn
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Toán Lớp 8: tìm gtnn x² – xy + y² – 2x – 2y
ko chép mạng nha . cảm ơn
Comments ( 2 )
x² – xy + y² – 2x – 2y
= x² – 2x + 1 + y² – 2y + 1 – xy – 2
= (x – 1)² + (y – 1)² – (xy + 2)
Mà (x – 1)² ≥ 0 ; (y – 1)² ≥ 0
Do đó x – 1 = 0 ; y – 1 = 0
<=> x = 1 ; y = 1
Vậy \text{GTNN} = -4 khi x = 1 ; y = 1
Gửi bạn:
Đặt: $Q=x^2-xy+y^2-2x-2y$
$⇒2Q=2(x^2-xy+y^2-2x-2y)$
$⇒2Q=2x^2-2xy+2y^2-4x-4y$
$2Q=(x^2-2xy+y^2)+(x^2-4x+4)+(y^2-4y+4)-8$
$2Q=(x-y)^2+(x-2)^2+(y-2)^2-8$
Ta có: $(x-y)^2+(x-2)^2+(y-2)^2-8≥-8(∀x)$
$⇒2Q≥-8$
$⇒Q≥-4$
Dấu $’=’$ xảy ra khi:
$x-y=0⇒x=y=2$
$x-2=0⇒x=2$
$y-2=0⇒y=2$
Vậy $Q_{MIN}=-4↔x=y=2$