The value of a and b such that the function
`f(x)={(-2sinx"," , -pi le x le -(pi)/(2)),(a sinx+b",", -(pi)/(2) lt x lt (pi)/(2)),(cosx",", (pi)/(2) le x le pi ):}` is continuous in `[-pi,pi]` are

Options

A. `-1,0`
B. `1,0`
C. 1,1
D. `-1,1`

Answer

D

Explanation

For continuity in `[-pi , pi]` , we must have
At `x = - ""(pi)/(2) , f (-""(pi)/(2)) = underset(x to (-""(pi)/(2))^(-)) ("lim") (-2 sin x )`
`= underset( x to (-""(pi)/(2))^(+)) (a sin x + b)`
`implies 2 = - a + b " " … (i) `
At `x = pi/2 ,`
`f((pi)/(2)) = underset(x to ((pi)/(2))^(-)) ("lim") ( asin x + b) = underset( x to ((pi)/(2))^(+)) ("lim") cos x`
`implies 0 = a + b " " .... (ii)`
On solving Eqs. (i) and (ii) we get , `a = -1 , b = 1`