Stationary inflection point ($f^{\prime }(a)=0$)

An inflection point of the graph of a differentiable real function at which its derivative is zero. It has a horizontal tangent and a change of concavity, but need not be a local maximum or minimum. For a three-times continuously differentiable function, $f^{\prime }(a)=f^{\prime \prime }(a)=0$ and $f^{\prime \prime \prime }(a)\ne 0$ suffice: the expansion $f^{\prime \prime }(a+h)=f^{\prime \prime \prime }(a)h+o(h)$ gives opposite signs on the two sides. The elementary example is $f(x)=x^3$ at zero. The word “saddle” is sometimes used for this one-dimensional situation in elementary examination papers, but the usual saddle point of a scalar function has a multivariable meaning; do not merge the two concepts.