Given `A={f:C[0,1]:int_(0)^(1)f(x)dx=1}` then minimum value of `int_(0)^(1)(1+x^(2))f^(2)(x)dx, epsilon A` is `k/(pi)` then `k` is

Answer

4

Explanation

By Schartz's inequality `1=|int_(0)^(1)f(x)dx"|=|int_(0)^(1)(f(x)sqrt(1+x^(2)dx))/(sqrt(1+x^(2)))|`
`1le sqrt(int_(0)^(1)(1+x^(2))f^(2)(x)dx)sqrt(int_(0)^(1)(dx)/(1+x^(2)))`
i.e. `1le sqrt(int_(0)^(1)(1+x^(2))f^(2)(x)dx.) (sqrt(pi))/2`
`:.` Required minimum is `4/(pi)` which is attained for `f(x)=4/(pi(1+x^(2)))`