= Saddle point of a scalar function
{title2=$\nabla f(x_*)=0$}
= Saddle points of a scalar function
{synonym}
A <saddle point of a scalar function> is a <stationary point> with nearby values both greater and less than its value. An indefinite <Hessian matrix> proves this by choosing directions with opposite signs of the quadratic term. A degenerate <Hessian matrix> may require higher-order terms: $x^2-y^4$ has a <saddle point of a scalar function> at the origin despite a zero quadratic coefficient in the $y$ direction. This is distinct from a <saddle equilibrium> of a <dynamical system>.
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