The maximum value of `cos xsin x+sqrt(sin^(2)x+sin^(2)((pi)/(6))}` is

Options

A. `sqrt(5)/(3)`
B. `sqrt((3)/(2))`
C. `sqrt((5)/(2))`
D. `sqrt(5)/(2)`

Answer

D

Explanation

Let `y=cosx{sinx+sqrt(sin^(2)x+sin^(2)((pi)/(6))]` is
`rArr y^(2)tan^(2)x-2ytan x+(y^(2)-(1)/(4))ge0`
For real `tan x`,
disc `ge0`
`4y^(2)-4y^(2)(y^(2)-(1)/(4))ge0`
`rArr 0ley^(2)le(5)/(4)`
`therefore` Maximum vlaue of `y=sqrt(5)/(2)`.