Tìm GTNN - KHÓ

T

transformers123

Áp dụng bđt Cauchy, ta có:
$M=(1+x^4)(1+y^4)+4(xy-1)(3xy-1)$
$\iff M \ge 2x^2.2y^2+4(3x^2y^2-4xy+1)$
$\iff M \ge 4x^2y^2+12x^2y^2-16xy+4$
$\iff M \ge 16x^2y^2-16xy+4$
$\iff M \ge 4(2xy-1)^2$
Vì $x+y=2$ nên $y=2-x$, thế vào, ta có:
$M \ge 4[2x(2-x)-1]^2$
$\iff M \ge 4(4x-2x^2-1)^2$
$\iff M \ge 4(2x^2-4x+2-1)^2$
$\iff M \ge 4[2(x-1)^2-1]^2 \ge 4(-1)^2 = 4$
Vậy $M_{Min}=4$ khi $\begin{cases}x^4=1\\y^4=1\\x+y=2\\x-1=0\end{cases} \rightarrow x=y=1$
 
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