1/ \(P=a^2+b^2+\frac{1}{a^2}+\frac{1}{b^2}\)
\(=\left(a^2+\frac{1}{16a^2}\right)+\left(b^2+\frac{1}{16b^2}\right)+\frac{15}{16}\left(\frac{1}{a^2}+\frac{1}{b^2}\right)\)
\(\ge\frac{1}{2}+\frac{1}{2}+\frac{15}{16}.\frac{2}{ab}\)
\(\ge1+\frac{15}{8}.\frac{1}{\frac{\left(a+b\right)^2}{4}}\le1+\frac{15}{8}.\frac{1}{\frac{1}{4}}=\frac{17}{2}\)
2/ \(\sqrt{x+4}+\sqrt{x-4}=2x-12+2\sqrt{x^2-16}\)
Đặt \(\hept{\begin{cases}\sqrt{x+4}=a\\\sqrt{x-4}=b\end{cases}}\)
\(\Rightarrow a+b=a^2+b^2-12+2ab\)
\(\Leftrightarrow\left(a+b\right)^2-\left(a+b\right)-12=0\)
\(\Leftrightarrow\orbr{\begin{cases}a+b=-3\left(l\right)\\a+b=4\end{cases}}\)
\(\Leftrightarrow\sqrt{x+4}+\sqrt{x-4}=4\)
Tới đây đơn giản rồi làm nốt nhé.