Solution
= Solution
Apply the <chain rule> to the phase-space function:
$$
\frac{dF}{dt}=\sum_i\left(F_{q_i}\dot q_i+F_{p_i}\dot p_i\right)
=\sum_i(F_{q_i}H_{p_i}-F_{p_i}H_{q_i})=[F,H].
$$
Thus $\boxed{[F,H]=0\Longrightarrow F\text{ is conserved}}$. If an explicitly time-dependent function is allowed, the correct equation is $dF/dt=\partial_tF+[F,H]$, so a vanishing bracket alone would not suffice. The source specifies $F(\mathbf q,\mathbf p)$, for which that extra term is absent.