$\underline{\mathbf{Answer:D}}$
$\underline{\mathbf{Solution:}}$
Let's say, $\mathrm x$ is a root then $x\mathrm{^2+1}$ should also be a root.
Thus, if $\mathrm {x_1}$ is the largest real root, then we should have $\mathrm{x_1 \ge x_1^2+1}$
$\mathrm{\Rightarrow x_1^2 -x_1 + 1 \le 0}$
But,
$\mathrm{x_1^2-x_1+1 = (x_1-\frac{1}{2})^2 + \frac{3}{4} > 0}$
which is a contradiction.
$\therefore \mathbf f$ has no real roots.
$\therefore \mathbf D$ is the correct answer.