 Pointwise optimal kernel bandwidth

For a second-order kernel density estimator, if $f(x)>0$ and $f^{\prime \prime }(x)\ne 0$, the leading pointwise asymptotic mean squared error is $R(K)f(x)/(nh)+{\mu }_2(K{)}^2{f}^{\prime \prime }(x{)}^2{h}^4/4$. Differentiation gives the displayed interior optimum. At a zero of $f$, the leading variance term vanishes and this two-term approximation has no positive interior minimizer; at a zero of $f^{\prime \prime }$, a higher-order bias expansion is needed.