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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $f \\colon U \\to \\mathbb{R}$ be [[differentiable|0135XZZQ8M]] at $a \\in U$.\nThe point $a$ is a saddle point of $f$ if $\\nabla f(a) = 0$ and $a$ is not a [[local extremum|04CPG9K1C4]] of $f$.\n\n## Remarks\n\nEquivalently, $\\nabla f(a) = 0$ and each ball around $a$ contains points $x$ and $y$ with $f(x) \u003c f(a) \u003c f(y)$.\nThe origin is a saddle point of $f(x, y) = x^2 - y^2$. Along the $x$-axis $f$ has a minimum at $0$, and along the $y$-axis it has a maximum.\nIf $f$ is of class $C^2$ and the [[Hessian matrix|0SCSQWXRA4]] at a critical point is indefinite, the point is a saddle point. This condition is not necessary: $f(x, y) = x^3$ has a saddle point at $(0, 0)$ with zero Hessian.\nIn one variable, a saddle point is a stationary point that is not an extremum, such as $0$ for $x \\mapsto x^3$.\n\n## Explanations\n\n### L1 human\n\nThe name comes from the shape of a horse saddle.\nIt curves up from front to back and curves down from side to side. So the centre is flat, but it is neither a top nor a bottom.\n",
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