Ta chứng minh được các bất đẳng thức bằng biến đổi tương đương và bất đẳng thức Cô-si:

\(x+y+z\le\sqrt{3\left(x^2+y^2+z^2\right)}\)

\(xy+yz+zx\ge3\sqrt[3]{\left(xyz\right)^2}\)

\(\Rightarrow\frac{xyz}{xy+yz+zx}\le\frac{\sqrt[3]{xyz}}{3}\)

Mà \(\sqrt[3]{xyz}\le\frac{x+y+z}{3}\le\frac{\sqrt{3\left(x^2+y^2+z^2\right)}}{3}\)

Vậy \(A\le\frac{\sqrt{3\left(x^2+y^2+z^2\right)}}{3}.\frac{\sqrt{3\left(x^2+y^2+z^2\right)}+\sqrt{x^2+y^2+z^2}}{x^2+y^2+z^2}\)

\(A\le\frac{\sqrt{3}\left(\sqrt{3}+1\right)}{3}=\frac{3+\sqrt{3}}{3}\)

Vì xyz=1\(\Rightarrow x^2\left(y+z\right)\ge2x^2\sqrt{yz}=2x\sqrt{x}\)

Tương tự \(y^2\left(z+x\right)\ge2y\sqrt{y};z^2=\left(x+y\right)\ge2z\sqrt{z}\)

\(\Rightarrow P\ge\frac{2x\sqrt{x}}{y\sqrt{y}+2z\sqrt{z}}+\frac{2y\sqrt{y}}{z\sqrt{z}+2x\sqrt{x}}+\frac{2z\sqrt{z}}{x\sqrt{x}+2y\sqrt{y}}\)

Đặt \(x\sqrt{x}+2y\sqrt{y}=a;y\sqrt{y}+2z\sqrt{z}=b;z\sqrt{z}+2x\sqrt{x}=c\)

\(\Rightarrow x\sqrt{x}=\frac{4c+a-2b}{9};y\sqrt{y}=\frac{4a+b-2c}{9};z\sqrt{z}=\frac{4b+c-2a}{9}\)

\(\Rightarrow P\ge\frac{2}{9}\left(\frac{4c+a-2b}{b}+\frac{4a+b-2c}{a}+\frac{4b+c-2a}{b}\right)\)

\(=\frac{2}{9}\text{ }\left[4\left(\frac{c}{b}+\frac{a}{c}+\frac{b}{a}\right)+\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)-6\right]\ge\frac{2}{9}\left(4.3+2-6\right)=2\)

Min P =2 khi và chỉ khi a=b=c khi va chỉ khi x=y=z=1

Dự đoán \(x=y=z=1\) ta tính được \(A=6+3\sqrt{2}\)

Ta sẽ c/m nó là GTLN của A

Thật vậy, ta cần chứng minh \(Σ\left(2+\sqrt{2}-2\sqrt{x}-\sqrt{1+x^2}\right)\ge0\)

\(\LeftrightarrowΣ\left(\frac{2\left(1-x\right)}{1+\sqrt{x}}+\frac{1-x^2}{\sqrt{2}+\sqrt{1+x^2}}\right)\ge0\)

\(\LeftrightarrowΣ\left(x-1\right)\left(1+\frac{1}{\sqrt{2}}-\frac{2}{1+\sqrt{x}}-\frac{x+1}{\sqrt{2}+\sqrt{1+x^2}}\right)+\left(1+\frac{1}{\sqrt{2}}\right)\left(3-x-y-z\right)\ge0\)

\(\LeftrightarrowΣ\left(x-1\right)^2\left(\frac{1}{\left(1+\sqrt{x}\right)^2}-\frac{x+1}{\sqrt{2}\left(\sqrt{2}+\sqrt{1+x^2}\right)\left(\sqrt{2}x+\sqrt{1+x^2}\right)}\right)+\left(1+\frac{1}{\sqrt{2}}\right)\left(3-x-y-z\right)\ge0\)

BĐT cuối đủ để chứng minh 

\(\sqrt{2}\left(\sqrt{2}+\sqrt{1+x^2}\right)\left(\sqrt{2}x+\sqrt{1+x^2}\right)\ge\left(x+1\right)\left(1+\sqrt{x}\right)^2\)

Đặt \(1+x=2k\sqrt{x}\). Hence, theo Cauchy-Schwarz:

\(\sqrt{2}\left(\sqrt{2}+\sqrt{1+x^2}\right)\left(\sqrt{2}x+\sqrt{1+x^2}\right)\)

\(=\sqrt{2}\left(\sqrt{2}+\frac{1}{\sqrt{2}}\sqrt{2\left(1+x^2\right)}\right)\left(\sqrt{2}x+\frac{1}{\sqrt{2}}\sqrt{2\left(1+x^2\right)}\right)\)

\(\ge\sqrt{2}\left(\sqrt{2}+\frac{x+1}{\sqrt{2}}\right)\left(\sqrt{2}x+\frac{x+1}{\sqrt{2}}\right)\)

\(=\frac{1}{\sqrt{2}}\left(x+3\right)\left(3x+1\right)=\frac{1}{\sqrt{2}}\left(3x^2+10x+3\right)\)

\(=\frac{1}{\sqrt{2}}\left(3\left(4k^2-2\right)x+10x\right)2\sqrt{2}x\left(3k^2+1\right)\)

Mặt khác \(\left(x+1\right)\left(1+\sqrt{x}\right)^2=\left(x+1\right)\left(x+1+2\sqrt{x}\right)\)

\(=2k\left(2k+2\right)x=4k\left(k+1\right)x\). Có nghĩa là ta cần phải c/m

\(3k^2+1\ge\sqrt{2}k\left(k+1\right)\Leftrightarrow\left(3-\sqrt{2}\right)k^2-2\sqrt{k}+1\ge0\)

Nó đúng theo AM-GM

\(\left(3-\sqrt{2}\right)k^2-\sqrt{2}k+1\ge\left(2\sqrt{3-\sqrt{2}}-\sqrt{2}\right)k\ge0\)

Hơi đẹp nhỉ nhưng xong r` đó :D

bunyakovsky:

\(\left(\sqrt{1+x^2}+\sqrt{2x}\right)^2\le2\left(x+1\right)^2\)

\(\Leftrightarrow\sqrt{1+x^2}+\sqrt{2}.\sqrt{x}\le\sqrt{2}\left(x+1\right)\) 

tương tự :phần còn lại + thêm với\(\left(2-\sqrt{2}\right)\left(x+y+z\right)\)

Ta có:

\(1.\sqrt{1+x^2}+1.\sqrt{2x}\le\sqrt{\left(1+1\right)\left(1+x^2+2x\right)}=\sqrt{2}\left(x+1\right)\)

Tương tự:

\(\sqrt{1+y^2}+\sqrt{2y}\le\sqrt{2}\left(y+1\right)\) ; \(\sqrt{1+z^2}+\sqrt{2z}\le\sqrt{2}\left(z+1\right)\)

Cộng vế:

\(P\le\sqrt{2}\left(x+y+z+3\right)+\left(2-\sqrt{2}\right)\left(x+y+z\right)\le\sqrt{2}\left(3+3\right)+\left(2-\sqrt{2}\right).3=6+3\sqrt{2}\)

\(P_{max}=6+3\sqrt{2}\) khi \(x=y=z=1\)