Find the largest open interval on which the function is increasing.
\[
f(x)=\frac{1}{x^{2}+1}
\]
A. $(-\infty, 1)$
B. $(1, \infty)$
C. $(-\infty, 0)$
D. $(0, \infty)$

Step 1 ：Given the function \(f(x)=\frac{1}{x^{2}+1}\), we need to find the largest open interval on which the function is increasing.

Step 2 ：To do this, we first find the derivative of the function, which is \(f'(x)=-\frac{2x}{(x^{2}+1)^{2}}\).

Step 3 ：We then set the derivative equal to zero to find the critical points. Solving \(-\frac{2x}{(x^{2}+1)^{2}}=0\), we find that the critical point is at \(x=0\).

Step 4 ：We now test the intervals \((-\infty, 0)\) and \((0, \infty)\) to see where the function is increasing. We do this by substituting a number from each interval into the derivative and checking the sign.

Step 5 ：For the interval \((-\infty, 0)\), we can choose \(x=-1\) as a test point. Substituting \(x=-1\) into the derivative, we find that \(f'(-1)>0\), so the function is increasing on this interval.

Step 6 ：For the interval \((0, \infty)\), we can choose \(x=1\) as a test point. Substituting \(x=1\) into the derivative, we find that \(f'(1)<0\), so the function is decreasing on this interval.

Step 7 ：Therefore, the largest open interval on which the function is increasing is \((-\infty, 0)\).

Step 8 ：Final Answer: \(\boxed{(-\infty, 0)}\)