The maximum ordinate of a point on the graph of the function f(x) = sin x( 1+ cos x ) is

Options

A. `(2+sqrt(3))/(4)`
B. `(3sqrt(3))/(4)`
C. 1
D. non of these

Answer

D

Explanation

We have
`f(x) =sin ""x+1/2 sin "" 2x`
`rArr f(x) = cos + cos^2 x and f(x) =- sin x-2 sin 2 x `
For local maximum or minimum we must have
f(x) =0
`rArr cos x- cos2x =0`
`rArr 2 cos^2 x + cos x-1 =0`
`rArr ( 2 cos x -1 ) (cos x -1 )=0`
`rArr cos x ""= 1/2 , -1 rArr x= pi/3 ,pi, (5pi)/3`
Now `f'((pi)/(3))=-sin""pi/3-2 sin"" j(2pi)/(3) lt 0 f' (pi)=0`
Thus f(x) attains a local maximum at `x=pi/3`
`therefore` Maximum ordinater `=f((pi)/(3))=sin ""pi/3(1+ cos""pi/3)`
`=sqrt(3)/2xx3/2=(3sqrt(3))/(4)`