Question

cho a ≥ 9 b≥4 c≥1 cmr:.....

\(ab\sqrt{c-1}+bc\sqrt{a-9}+ca\sqrt{b-4}\le\dfrac{11abc}{12}\)

Answer 1

Áp dụng bđt AM-GM cho 2 số không âm ta có:\(ab\sqrt{c-1}+bc\sqrt{a-9}+ca\sqrt{b-4}\)

\(=ab\sqrt{1.\left(c-1\right)}+\dfrac{bc\sqrt{9\cdot\left(a-9\right)}}{3}+\dfrac{ca\sqrt{4.\left(b-4\right)}}{2}\)\(\le ab.\dfrac{1+\left(c-1\right)}{2}+bc.\dfrac{9+\left(a-9\right)}{6}+ca.\dfrac{4+\left(b-4\right)}{4}=abc\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{4}\right)=\dfrac{11abc}{12}\left(đpcm\right)\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}1=c-1\\9=a-9\\4=b-4\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}c=2\\a=18\\b=8\end{matrix}\right.\)