= Solution
Use disjoint small neighborhoods of the finitely many nondegenerate stationary points $t_j$, apply the <stationary phase> formula in each, and <integration by parts> elsewhere. The leading contributions add:
$$
\boxed{I(\lambda)=\sum_j f(t_j)\sqrt{\frac{2\pi}{\lambda|g''(t_j)|}}e^{i\lambda g(t_j)+i\pi\operatorname{sgn}(g''(t_j))/4}+O(\lambda^{-1}).}
$$
Interference between these oscillatory contributions can cancel the leading coefficient at particular values of $\lambda$.
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