MT【287】余弦的线性组合

(2017北大特优)在$\Delta ABC$中,$cos A+\sqrt{2}cos B+\sqrt{2}cos C$的最大值____

解答 :2
$cos A+\sqrt{2}cos B+\sqrt{2}cos C$

$=cos A+2\sqrt{2}cos\dfrac{B+C}{2}cos\dfrac{B-C}{2}\le1-2sin^2\dfrac{A}{2}+2\sqrt{2}sin\dfrac{A}{2}\le2$

提示:$p\cos\alpha+q\cos\beta+r\cos\gamma\le\dfrac{1}{2}\sum\limits_{cyc}\dfrac{qr}{p}$
嵌入不等式:$$2yzcosA+2zxcosB+2xycosC\le x^2+y^2+z^2$$中令$(x,y,z)=(\sqrt{\dfrac{qr}{p}},\sqrt{\dfrac{rp}{q}},\sqrt{\dfrac{pq}{r}})$即得.