Suppose `f` is a real-valued differentiable function defined on `[1,oo]` with `f(x1)=1.` Moreover, suppose that `f` satisfies `f^(prime)(x)=1/(x^2+f^2(x))S howt h a tf(x)<1+pi/4AAxgeq1.`

Explanation

Here `f'(x)=1/(x^(2)+(f(x))^(2)) gt 0AA x ge 1`
Thus, `f(x)` is an increasing function `Axxge1`.
Given `f(1)=1`. So `f(x)ge1 AAxge1`.
Hence `f'(x)le1/(1+x^(2)) AA xge1`
or `int_(1)^(x)f'(x)dx le int_(1)^(x)1/(1+x^(2))dx`
or `f(x)=f(1)le tan^(-1)x-tan^(-1)1`
or `f(x)le tan^(-1)x +1-(pi)/4`
or `f(x) lt (pi)/2 +1-(pi)/4` (as `tan^(-1) x lt (pi)/2 AAxge1`)
i.e. `f(x) lt 1+(pi)/4AAxge1`