Question

[camcon]

Cho \(a\ge1;b\ge9;c\ge16\) và a.b.c = 1152

Tìm GTLN của biểu thức \(P=bc\sqrt{a-1}+ca\sqrt{b-9}+ab\sqrt{c-16}\)

Answer 1

[Nguyễn Việt Lâm]

\(P=bc.1.\sqrt{a-1}+\dfrac{ca}{3}.3.\sqrt{b-9}+\dfrac{ab}{4}.4.\sqrt{c-16}\)

\(P\le\dfrac{bc}{2}\left(1+a-1\right)+\dfrac{ca}{6}\left(9+b-9\right)+\dfrac{ab}{8}\left(16+c-16\right)\)

\(\Rightarrow P\le\dfrac{abc}{2}+\dfrac{abc}{6}+\dfrac{abc}{8}=912\)

\(P_{max}=912\)  khi \(\left(a;b;c\right)=\left(2;18;32\right)\)