Ta có:\(a+b\ge2\sqrt{ab}\Rightarrow a-2\sqrt{ab}+b\ge0\Rightarrow\left(\sqrt{a}\right)^2-2\sqrt{ab}+\left(\sqrt{b}\right)^2\ge0\Rightarrow\left(\sqrt{a}+\sqrt{b}\right)^2\ge0\left(LĐ\right)\)Dấu "=" xảy ra <=> a = b

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

\(P=\left[-\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]\left(-\frac{\sqrt{x}}{2}+\frac{1}{2\sqrt{x}}\right)^2\)

\(P=\left[-\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]\left(\frac{1}{4x}+\frac{1}{4}-\frac{1}{2}\right)\)

\(P=-\frac{4\sqrt{x}.\left(\frac{1}{4x}-\frac{1}{2}+\frac{x}{4}\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)

\(P=-\frac{4.\frac{x^2-2x+1}{4x}.\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\)

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

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

\(P=-\frac{\sqrt{x}.\left(x-1\right)}{x}\)

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VÌ \(c\le3a\)

=> \(4\ge\left(a+2b\right)\left(\frac{1}{b}+\frac{3}{a}\right)\)

<=> \(\frac{5}{3}\ge\left(\frac{a}{b}+\frac{b}{a}\right)-\frac{b}{3a}\ge2-\frac{b}{3a}\)

=> \(\frac{b}{a}\ge1\)=> \(b\ge a\)

Khi đó 

\(\frac{a^2+2b^2}{ac}\ge\frac{3a^2}{a.3a}=1\)(ĐPCM)

Dấu bằng xảy ra khi \(c=3a=3b\)

pt \(\Leftrightarrow\)\(x^4+2x^2y^2+y^4=2y^2-x^2+3\)

\(\Leftrightarrow\)\(\left(x^2+y^2\right)^2-2\left(x^2+y^2\right)+1=-3x^2+4\)

\(\Leftrightarrow\)\(\left(x^2+y^2-1\right)^2=-3x^2+4\le4\)

\(\Rightarrow\)\(-1\le x^2+y^2\le3\)