设n（n≥2）是给定的整数,x1,x2,…,xn（n是下标）是实数,则sinx1·cosx2+sinx2·cosx3+…+sinxn·cosx1的最大值是n/2sinx1·cosx2≤sin^2x1+cos^2x2怎么证？

Question

设n（n≥2）是给定的整数,x1,x2,…,xn（n是下标）是实数,则sinx1·cosx2+sinx2·cosx3+…+sinxn·cosx1 的最大值是________
n/2
sinx1·cosx2≤sin^2x1+cos^2x2 怎么证？

Answer 1

因为,sin^2 x1+cos^2 x2-2sinx1·cosx2>=0
所以,sinx1·cosx2≤（sin^2 x1+cos^2 x2）/2
sinx1·cosx2+sinx2·cosx3+…+sinxn·cosx1
<=[(sin^2x1+cos^2x2)+(sin^2x2+cos^2x3)+…+(sin^2xn+cos^2x1) ]/2
<=（sin^2x1+cos^2x1+sin^2x2+cos^2x2+…………sin^2xn+cos^2xn）/2
<=(1+1+1+…………+1)/2
<=n/2