The greatest value of `(2sin theta+3cos theta+4)^(3).(6-2sin theta-3cos theta)^(2)`, as `theta in R`, is

Options

A. 2345
B. 3456
C. 1234
D. 4567

Answer

B

Explanation

Clearly, `2 sin theta + 3 cos theta + 4 gt 0 AA theta in R`
and `6-2 sin theta -3 cos theta gt 0 AA x in R`
Put, `2 sin theta+3 cos theta + 4 = x` and `6-2 sin theta -3 cos theta =y`
Now, `x+y=10`
Also, A.M. ge G.M. (for positive numbers)
`therefore ((x)/(3)+(x)/(3)+(x)/(3)+(y)/(2)+(y)/(2))/(5)ge ((x^(3))/(27).(y^(2))/(4))^((1)/(5))`
`therefore (2 sin theta + 3 cos theta + 4)^(3)(6-2 sin theta - 3 cos theta)^(2)le (32)(27)(4)=3456`