= Solution
Let $Q_k$ be a vibrational <normal mode coordinate> with symmetry $\Gamma_k$. The harmonic vibrational ground state is totally symmetric; the one-quantum state transforms as $Q_k$. Integration over a symmetry-invariant domain annihilates any integrand lacking the <totally symmetric representation>. Thus a fundamental requires
$$
\Gamma(v'=1_k)\otimes\Gamma(\mu_\alpha)\otimes\Gamma(v''=0)
\supset A_1
$$
for some Cartesian component $\alpha$. For the one-dimensional irreducible representations used here, this means $\Gamma_k=\Gamma(x)$, $\Gamma(y)$ or $\Gamma(z)$. Equivalently, expanding the <electric dipole moment> gives
$$
\mu_\alpha=\mu_{\alpha,0}+\left(\frac{\partial\mu_\alpha}{\partial Q_k}\right)_0Q_k+\cdots;
$$
the derivative can be nonzero only if $Q_k$ and $\mu_\alpha$ have compatible symmetry. This is the symmetry criterion for <vibrational infrared activity>; it permits a transition without guaranteeing a large intensity.
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