Consider the function
\[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\]
where $\log _{e}(x)$ denotes the natural logarithm of $x$.
Which of the following is true:
- The function $f(y)$ is non-positive for all $y \geq 1$.
- The function $f(y)$ first increases and then decreases with $y$ for $y \geq 1$.
- The function $f(y)$ first decreases and then increases with $y$ for $y \geq 1$.
- The function $f(y)$ oscillates infinitely often between negative and positive value for $y \geq 1$.
- The derivative of function $f(y)$ does not exist at $y=1$.