If `f(x)={(ax+1",", x le (pi)/(2)),(sinx+b",", x gt (pi)/(2)):}` is continuous, then

Options

A. `a=1, b= 0`
B. `a=b(pi)/(2)+1`
C. `b=(api)/(2)`
D. `a=b=(pi)/(2)`

Answer

C

Explanation

Given, `f(x) ={(ax+1",",x le (pi)/(2)),(sinx +b",", x gt (pi)/(2) ):}`,
Since f(x) is continuous at `x=(pi)/(2)`
`therefore lim_( x to (pi^(-))/(2)) f(x) = lim_(x to (pi^(+))/(2))f(x)`
`rArr =lim_(x to (pi^(-))/(2)) (ax+1)=lim_(x to (pi^(+))/(2) ) sin x +b`
` rArr a * (pi)/(2) +1 = 1+b rArr b =(a pi)/(2)`