Consider a real valued continuous function `f` such that `f(x)=sinx + int_(-pi//2)^(pi//2) (sinx+t(f(t))dt`. If `M` and `m` are maximum and minimum values of function `f`, then the value of `M//m` is____________.

Answer

3

Explanation

We have
`f(x)=sinx+int_(-pi//2)^(pi//2) (sinx+tf(t))dt=sinx+pisinx+int_(-pi//2)^(pi//2)tf(t)dt`
`:.f(x)=(pi+1)sinx+A` …..1
Now `A=int_(-pi//2)^(pi//2) t((pi+1)sint+A)dt=2(pi+1)(int_(0)^(pi//2) underset(("By part")) underset((I), (II)) (t sin t dt))`
`:.A=2(pi+1)`
Hence `f(x)=(pi+1)sinx+2(pi+1)`
Therefore `f_("max")=3(pi+1)=M`
and `f_("min")=(pi+1)=m`
`:.M/m=3`