求$sqrt{dfrac{5}{4}-sin x}+2sqrt{dfrac{9}{4}+cos x-sin x}$的最小值.
提示:
$sqrt{dfrac{5}{4}-sin x}+2sqrt{dfrac{9}{4}+cos x-sin x}$
$=sqrt{(dfrac{1}{2}cos x)^2+(1-dfrac{1}{2}sin x)^2}+2sqrt{(dfrac{1}{2}cos x+1)^2+(dfrac{1}{2}sin x-1)^2}$
令$A(-1,1),B(0,1),D(0,dfrac{1}{4}),C(dfrac{1}{2}cos x,dfrac{1}{2}sin x)$
则由内外圆知识$|BC|+2|AC|=2(|DC|+|AC|)ge 2AD=dfrac{5}{2}$