14. If the function $f(x)$ defined on the set $A$ satisfies: for any $x_{1}, x_{2} \in A$, we have
$$
f\left(\frac{x_{1}+x_{2}}{2}\right) \leqslant \frac{1}{2}\left[f\left(x_{1}\right)+f\left(x_{2}\right)\right],
$$

then the function $f(x)$ is called a concave function on $A$.
(1) Determine whether $f(x)=3 x^{2}+x$ is a concave function on $\mathbf{R}$.
(2) If the function $f(x)=m x^{2}+x$ is a concave function on $\mathbf{R}$, find the range of real numbers for $m$.