Question

Bài 1 : cho x, y >0 và x²+y²=1. Tìm GTNN của \(P=\left(1+x\right)\cdot\left(1+\dfrac{1}{y}\right)+\left(1+y\right)\cdot\left(1+\dfrac{1}{x}\right)\)

Bài 2 : cho a, b, c > 0. CMR

\(\dfrac{1}{a+3b}+\dfrac{1}{b+3c}+\dfrac{1}{c+3a}>=\dfrac{1}{2a+b+c}+\dfrac{1}{2b+a+c}+\dfrac{1}{2c+a+b}\)

Bài 3 : cho a, b, c, d >0. CMR

\(\dfrac{a+c}{a+b}+\dfrac{b+d}{b+c}+\dfrac{c+a}{c+d}+\dfrac{d+b}{d+a}>=4\)

Answer 1

Bài 1:

\(P=(x+1)\left(1+\frac{1}{y}\right)+(y+1)\left(1+\frac{1}{x}\right)\)

\(=2+x+y+\frac{x}{y}+\frac{y}{x}+\frac{1}{x}+\frac{1}{y}\)

Áp dụng BĐT Cô-si:

\(\frac{x}{y}+\frac{y}{x}\geq 2\)

\(x+\frac{1}{2x}\geq 2\sqrt{\frac{1}{2}}=\sqrt{2}\)

\(y+\frac{1}{2y}\geq 2\sqrt{\frac{1}{2}}=\sqrt{2}\)

Áp dụng BĐT SVac-xơ kết hợp với Cô-si:

\(\frac{1}{2x}+\frac{1}{2y}\geq \frac{4}{2x+2y}=\frac{2}{x+y}\geq \frac{2}{\sqrt{2(x^2+y^2)}}=\frac{2}{\sqrt{2}}=\sqrt{2}\)

Cộng các BĐT trên :

\(\Rightarrow P\geq 2+2+\sqrt{2}+\sqrt{2}+\sqrt{2}=4+3\sqrt{2}\)

Vậy \(P_{\min}=4+3\sqrt{2}\Leftrightarrow a=b=\frac{1}{\sqrt{2}}\)

Answer 2

Bài 2:

Áp dụng BĐT Svac-xơ:

\(\frac{1}{a+3b}+\frac{1}{b+a+2c}\geq \frac{4}{2a+4b+2c}=\frac{2}{a+2b+c}\)

\(\frac{1}{b+3c}+\frac{1}{b+c+2a}\geq \frac{4}{2b+4c+2a}=\frac{2}{b+2c+a}\)

\(\frac{1}{c+3a}+\frac{1}{c+a+2b}\geq \frac{4}{2c+4a+2b}=\frac{2}{c+2a+b}\)

Cộng theo vế và rút gọn :

\(\Rightarrow \frac{1}{a+3b}+\frac{1}{b+3c}+\frac{1}{c+3a}\geq \frac{1}{2a+b+c}+\frac{1}{2b+c+a}+\frac{1}{2c+a+b}\) (đpcm)

Dấu bằng xảy ra khi $a=b=c$

Answer 3

Bài 3:

Áp dụng BĐT Svacxo:

\(\frac{1}{a+b}+\frac{1}{c+d}\geq \frac{4}{a+b+c+d}\)

\(\Rightarrow \frac{a+c}{a+b}+\frac{a+c}{c+d}\geq \frac{4(a+c)}{a+b+c+d}(1)\)

\(\frac{1}{b+c}+\frac{1}{d+a}\geq \frac{4}{b+c+d+a}\)

\(\Rightarrow \frac{b+d}{b+c}+\frac{b+d}{d+a}\geq \frac{4(b+d)}{a+b+c+d}(2)\)

Từ \((1);(2)\Rightarrow \frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\geq \frac{4(a+c+b+d)}{a+b+c+d}=4\)

Dấu bằng xảy ra khi \(a=b=c=d\)