= Solution
Near the nondegenerate stationary point, write $g(t)=g(t_0)+g''(t_0)(t-t_0)^2/2+O((t-t_0)^3)$ and replace $f(t)$ by $f(t_0)$ to leading order. Scaling $t-t_0$ by $\lambda^{-1/2}$ reduces the local integral to the Fresnel integral, while the remaining nonstationary region contributes $O(\lambda^{-1})$. The <stationary phase> result is
$$
\boxed{I(\lambda)=f(t_0)\sqrt{\frac{2\pi}{\lambda|g''(t_0)|}}\exp\left(i\lambda g(t_0)+\frac{i\pi}4\operatorname{sgn}g''(t_0)\right)+O(\lambda^{-1}).}
$$
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