= Stationary inflection point
{title2=$f'(a)=0$}
= Stationary point of inflexion
{synonym}
= Stationary point of inflection
{synonym}
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>, $f'(a)=f''(a)=0$ and $f'''(a)\ne0$ suffice: the expansion $f''(a+h)=f'''(a)h+o(h)$ 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.
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