\(P = bc\sqrt {a - 1} + ca\sqrt {b - 9} + ab\sqrt {c - 16} \)

\( \Leftrightarrow \frac{P} = \frac{{\sqrt {a - 1} }}{a} + \frac{{\sqrt {b - 9} }}{b} + \frac{{\sqrt {c - 16} }}{c}\)

\( \Leftrightarrow \frac{P} = \frac{{\sqrt {a - 1} }}{a} + \frac{{\sqrt {b - 9} }}{b} + \frac{{\sqrt {c - 16} }}{c}\)

Áp dụng BĐT AM - GM:

\(\left\{ \begin{array}{l}2\sqrt {a - 1} \le a - 1 + 1 = a\\2\sqrt {9\left( {b - 9} \right)} \le b - 9 + 9 = b\\2\sqrt {16\left( {c - 16} \right)} \le c - 16 + 16 = c\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\frac{{\sqrt {a - 1} }}{a} \le \frac{1}{2}\\\frac{{\sqrt {b - 9} }}{b} \le \frac{1}{6}\\\frac{{\sqrt {c - 16} }}{c} \le \frac{1}{8}\end{array} \right.\)

Khi đó \(\frac{P} = \frac{{\sqrt {a - 1} }}{a} + \frac{{\sqrt {b - 9} }}{b} + \frac{{\sqrt {c - 16} }}{c} \le \frac{1}{2} + \frac{1}{6} + \frac{1}{8} = \frac\)

Û P ≤ 912

Dấu “=” xảy ra \( \Leftrightarrow \left\{ \begin{array}{l}a = 1\\b - 9 = 9\\c - 16 = 16\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}a = 1\\b = 18\\c = 32\end{array} \right.\)

Vậy GTLN của P là 912 khi (a; b; c) = (1; 18; 32).