If `f(x) = A sin ((pix)/(2)) + B` and `f'((1)/(2)) = sqrt(2)` and `int_(0)^(1)f(x) = (2A)/(pi)`, then what is the value of B ?

Options

A. `2/pi`
B. `4/pi`
C. `0`
D. `1`

Answer

C

Explanation

Given function `f(x) = A sin((pix)/(2)) + B`
Differentiating w.r.t.x
`f'(x) = A cos ((pix)/(2))pi/2`
`f'((1)/(2)) = sqrt(2) = A (cos'(pi)/(4)) (pi)/(2) = A. 1/(sqrt(2)). (pi)/(2)`.
`rArr A = ((sqrt(2) xx sqrt(2)) xx 2)/(pi) = 4/(pi)`
`rArr int_(0)^(1){A sin ((pix)/(2)) + B} dx = (2xx4)/(pi^(2))`
`rArr [-A cos '(pix)/(2).(2)/(x)+Bx]_(0)^(1) = (8)/(pi^(2))`
`rArr - (4)/(pi).(2)/(pi) cos'(pi)/(2) + B + (4)/(pi).(2)/(pi) cos theta = (8)/(pi^(2))`
`rArr B + (8)/(pi^(2)) = (8)/(pi^(2)) rArr B = 0`