If `f:R to R ` given by
`f(x)={(2cosx"," , "if", x le -(pi)/(2)),(a sin x+b",","if", -(pi)/(2) lt x lt (pi)/(2)),(1+cos^(2)x",","if",x ge(pi)/(2)):}` is a continuous function on R, then (a, b) is equal to

Options

A. `(1//2,1//2)`
B. `(0,-1)`
C. `(0,2)`
D. `(1,0)`

Answer

A

Explanation

`LHL=lim_(h to 0) f(-(pi)/(2)-h)=lim_(h to 0) 2 cos (-(pi)/(2)-h)=0`
`RHL=lim_(h to 0) f(-(pi)/(2) +h)=lim_(h to 0) a sin (-(pi)/(2)+h)+b`
`=-a+b`
Since, function is continuous.
`therefore RHL=LHL rArr a =b`
From the given options only (a) i.e. `((1)/(2),(1)/(2))` satisfies this condition.