\(\Leftrightarrow log_2\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=4\)
\(\Leftrightarrow\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=16\)
Đặt \(\left\{{}\begin{matrix}x+\sqrt{x^2+1}=a>0\\y+\sqrt{y^2+1}=b>0\end{matrix}\right.\) \(\Rightarrow ab=16\)
\(\sqrt{x^2+1}=a-x\Rightarrow x^2+1=a^2-2ax+x^2\)
\(\Rightarrow2ax=a^2-1\Rightarrow x=\dfrac{a^2-1}{2a}\)
Tương tự: \(y=\dfrac{b^2-1}{2b}\)
\(\Rightarrow P=x+y=\dfrac{a^2-1}{2a}+\dfrac{b^2-1}{2b}=\dfrac{a+b}{2}-\left(\dfrac{1}{2a}+\dfrac{1}{2b}\right)\)
\(=\dfrac{a+b}{2}-\dfrac{a+b}{2ab}=\dfrac{a+b}{2}-\dfrac{a+b}{32}=\dfrac{15}{32}\left(a+b\right)\ge\dfrac{15}{32}.2\sqrt{ab}=\dfrac{15}{4}\)
\(P_{min}=\dfrac{15}{4}\)
