Cubic convergence of Newton optimization for an even quartic ($f(x)=x^2/2+x^4/4$)

For the strictly convex polynomial $f(x)=x^2/2+x^4/4$, applying Newton root-finding iteration to $f^{\prime }(x)=x+x^3$ givesThe unique minimum is at zero. For any sufficiently small nonzero initial point, the errors satisfy $|x_{k+1}|/|x_k{|}^3\to 2$, so the order is cubic, not merely quadratic. This provides a simple counterexample to treating the usual quadratic convergence theorem as an upper bound on possible orders. In general, the quadratic coefficient for Newton optimization near a nondegenerate stationary point is $f^{(3)}(x_{\ast })/(2f^{\prime \prime }(x_{\ast }))$; when it vanishes, a higher order is possible.