[Toán 8] Tính GTNN bằng bất đẳng thức Bunhi

I

iceghost

$M=x^{2}+y^{2}-xy-x+y+1 \\
\implies 2M = 2x^2+2y^2-2xy-2x+2y+2 \\
=(x^2-2xy+y^2)+(x^2-2x+1)+(y^2+2y+1) \\
=(x-y)^2+(x-1)^2+(y+1)^2 \\
=(x-y)^2+(1-x)^2+(y+1)^2$

Theo bđt Bunhiacopxki ta có :
$(1^2+1^2+1^2)[(x-y)^2+(1-x)^2+(y+1)^2] \ge [1.(x-y)+1.(1-x)+1.(y+1)]^2 = 2^2 = 4 \\
\iff 3[(x-y)^2+(1-x)^2+(y+1)^2] \ge 4 \\
\iff (x-y)^2+(1-x)^2+(y+1)^2 = 2M \ge \dfrac43 \\
\implies M \ge \dfrac23 \\
\implies Min_M = \dfrac23$
Dấu '$=$' xảy ra khi $\dfrac{x-y}1=\dfrac{x-1}1=\dfrac{y+1}1 \iff x=3 ; y=1$
 
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