Math 2.0

Real functions Definition

Saddle point

Builds on Local maximum and minimumGradientDifferentiable function of several variables

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let f ⁣:U→ℝf \colon U \to \mathbb{R} be differentiable at a∈Ua \in U. The point aa is a saddle point of ff if ∇f(a)=0\nabla f(a) = 0 and aa is not a local extremum of ff.

Remarks

Equivalently, ∇f(a)=0\nabla f(a) = 0 and each ball around aa contains points xx and yy with f(x)<f(a)<f(y)f(x) < f(a) < f(y). The origin is a saddle point of f(x,y)=x2−y2f(x, y) = x^2 - y^2. Along the xx-axis ff has a minimum at 00, and along the yy-axis it has a maximum. If ff is of class C2C^2 and the Hessian matrix at a critical point is indefinite, the point is a saddle point. This condition is not necessary: f(x,y)=x3f(x, y) = x^3 has a saddle point at (0,0)(0, 0) with zero Hessian. In one variable, a saddle point is a stationary point that is not an extremum, such as 00 for x↦x3x \mapsto x^3.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Saddle_point, lead section
  2. Web https://en.wikipedia.org/wiki/Saddle_point, Mathematical discussion
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