= Solution
In the position representation, $\hat x$ multiplies by $x$ and $\hat p_x=-i\hbar\,\mathrm d/\mathrm dx$. Acting on an arbitrary differentiable test function $f$ shows the <commutator> explicitly:
$$
\begin{aligned}
[\hat x,\hat p_x]f
&=-i\hbar x f'+i\hbar\frac{\mathrm d(xf)}{\mathrm dx}\\
&=-i\hbar xf'+i\hbar(f+xf')=i\hbar f.
\end{aligned}
$$
Therefore $[\hat x,\hat p_x]=i\hbar\hat 1$, whose <expectation value> is $i\hbar$ in every normalized state in the relevant domains. The <Robertson uncertainty principle> gives
$$
\boxed{\Delta x\,\Delta p_x\ge\frac{\hbar}{2}.}
$$
This is a lower bound, not a claim that every state saturates it.
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