If `f(x)=[{:(mx+1,if x le (pi)/(2)),(sinx+n,ifxgt(pi)/(2)):}` is continuous at `x = (pi)/(2)`, then

Options

A. `m = 1, n = 0`
B. `m = (n pi)/(2)+1`
C. `n = (mpi)/(2)`
D. `m = n = (pi)/(2)`

Answer

C

Explanation

We have, `f(x)=[{:(mx+1,if x le (pi)/(2)),(sinx+n,ifxgt(pi)/(2)):}` is continuous at `x = (pi)/(2)`,
`:. LHL = underset(xrarr(pi^(-))/(2))(lim)(mx+1)=underset(hrarr0)(lim)[m(pi/2-h)+1]=(mpi)/(2)+1`
and `RHL = underset(xrarr(pi^(+))/(2))(lim)(sinx+n)=underset(hrarr0)(lim)[sin(pi/2+h)+n]`
`= underset(hrarr0)(lim)cosh+n=1+n`
`:. LHL = RHL` , [to be continuous at `x = (pi)/(2)`]
`rArr m.(pi)/(2) + 1 = n + 1`
`:. n = m.(pi)/(2)`