10. A die is thrown twice, and the numbers facing up are $m$ and $n$ respectively. Then the probability that the function $y=\frac{2}{3} m x^{3}-n x+1$ is an increasing function on $[1,+\infty)$ is ( ).
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{5}{6}$
10. D.

Notice that $f^{\prime}(x)=2 m x^{2}-n \geqslant 0$ always holds on $[1,+\infty]$, which means $2 x^{2} \geqslant \frac{n}{m}$ always holds, or equivalently, $2 m \geqslant n$. However, the pairs that satisfy $2 m < n$ are $(1,3)$, $(1,4)$, $(1,5)$, $(1,6)$, $(2,5)$, and $(2,6)$, a total of six. Therefore, $1-\frac{6}{36}=\frac{5}{6}$.