\(P=\dfrac{\sqrt{a-2}}{a}+\dfrac{\sqrt[3]{b-3}}{b}+\dfrac{\sqrt[4]{c-6}}{c}\)
\(=\dfrac{\sqrt{\left(a-2\right).2}}{a\sqrt{2}}+\dfrac{\sqrt[3]{\left(b-3\right).\dfrac{3}{2}.\dfrac{3}{2}}}{b\sqrt[3]{\dfrac{9}{4}}}+\dfrac{\sqrt[4]{\left(c-6\right).2.2.2}}{c\sqrt[3]{8}}\)
\(\le\dfrac{a-2+2}{2a\sqrt{2}}+\dfrac{b-3+\dfrac{3}{2}+\dfrac{3}{2}}{3b\sqrt[3]{\dfrac{9}{4}}}+\dfrac{c-6+2+2+2}{4c\sqrt[4]{8}}\)
\(=\dfrac{a}{2a\sqrt{2}}+\dfrac{b}{3b\sqrt[3]{\dfrac{9}{4}}}+\dfrac{c}{4c\sqrt[4]{8}}=\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt[3]{\dfrac{9}{4}}}+\dfrac{1}{4\sqrt[4]{8}}\)
Vậy \(P_{max}=\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt[3]{\dfrac{9}{4}}}+\dfrac{1}{4\sqrt[4]{8}}\)
Đẳng thức xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}a-2=2\\b-3=\dfrac{3}{2}\\c-6=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=4\\b=\dfrac{9}{2}\\c=8\end{matrix}\right.\)
\(P=\dfrac{bc\sqrt{a-2}+ac\sqrt[3]{b-3}+ab\sqrt[4]{c-6}}{abc}\)
\(=\dfrac{\sqrt{a-2}}{a}+\dfrac{\sqrt[3]{b-3}}{b}+\dfrac{\sqrt[4]{c-6}}{c}\)
Áp dụng BĐT AM-GM ta có:
\(=\dfrac{\sqrt{2\left(a-2\right)}}{\sqrt{2}a}+\dfrac{\sqrt[3]{2\left(b-3\right)}}{\sqrt[3]{2}b}+\dfrac{\sqrt[4]{2\left(c-6\right)}}{\sqrt[4]{2}c}\)
\(\le\dfrac{\dfrac{2+a-2}{2}}{\sqrt{2}a}+\dfrac{\dfrac{2+b-3+1}{3}}{\sqrt[3]{2}b}+\dfrac{\dfrac{2+c-6+1+1+1+1}{4}}{\sqrt[4]{2}c}\)
\(=\dfrac{\dfrac{a}{2}}{\sqrt{2}a}+\dfrac{\dfrac{b}{3}}{\sqrt[3]{2}b}+\dfrac{\dfrac{c}{4}}{\sqrt[4]{2}c}=\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt[3]{2}}+\dfrac{1}{4\sqrt[4]{2}}\)