tìm GTNN của $(1+x)(1+\frac{1}{y})+(1+y)(1+\frac{1}{x})$ thỏa $x^{2}+y^{2}=1$
tìm GTNN
Started By uchiha hitachi, 15-05-2017 - 21:06
#1
Posted 15-05-2017 - 21:06
#2
Posted 15-05-2017 - 23:25
tìm GTNN của $(1+x)(1+\frac{1}{y})+(1+y)(1+\frac{1}{x})$ thỏa $x^{2}+y^{2}=1$
Đặt $P=(1+x)(1+\frac{1}{y})+(1+y)(1+\frac{1}{x})$
Ta có:
$P=2+x+y+(\frac{1}{x}+\frac{1}{y})+(\frac{x}{y}+\frac{y}{x})\geq 4+x+y+\frac{4}{x+y}$
Mặt khác: $x+y\leq \sqrt{2(x^{2}+y^{2})}=\sqrt{2}$
Do đó:
$P=4+(x+y)+\frac{2}{x+y}+\frac{2}{x+y}\geq 4+3\sqrt{2}$
Dấu $=$ xảy ra khi $x=y=\frac{1}{\sqrt{2}}$
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