Solution
= Solution
Write $X=\mu+\sigma Z$ with $Z$ standard normal and set $g(z)=f(\mu+\sigma z)$. The <Gaussian Poincaré inequality> and the <chain rule> give
$$
\operatorname{Var}(f(X))
=\operatorname{Var}(g(Z))
\leq\mathbb E[g'(Z)^2]
=\sigma^2\mathbb E[f'(X)^2].
$$
Hence $N(\mu,\sigma^2)$ satisfies a $\sigma$-Poincaré inequality in the question's convention, where the constant is squared on the right-hand side.