Saddle Point¶

Definition¶

Let $f\colon A\subseteq\mathbb{R}^n\to\mathbb{R}$ and let $\boldsymbol{a}\in A$ $A$ . The point $\boldsymbol{a}$ is a saddle point if it is a stationary point but is neither a local maximum nor a local minimum.

Equivalently, every neighbourhood of $\boldsymbol{a}$ contains points where $f$ is larger than $f(\boldsymbol{a})$ and points where $f$ is smaller than $f(\boldsymbol{a})$ .

Interpretation¶

A saddle point is a flat-looking candidate where the function rises in some directions and falls in others: the tangent plane is horizontal, yet the point is no extremum.

Surface of f(x) = x1^2 - x2^2 and its level curves, which cross at the origin: f is positive in two opposite sectors and negative in the other two.

The figure shows $f(\boldsymbol{x})=x_1^2-x_2^2$ and its level curves. The level set $f=0$ consists of the two dashed lines $x_2=\pm x_1$ $\pm x_{1}$ , crossing at the origin: $f>0$ in the left and right sectors and $f<0$ in the upper and lower ones. At the origin $\boldsymbol{\nabla} f(\boldsymbol{0})=\boldsymbol{0}$ and $\boldsymbol{H}_f(\boldsymbol{0})=\operatorname{diag}(2,-2)$ is indefinite, so every neighbourhood contains both higher and lower values.

At the nondegenerate saddles shown here (where the Hessian is invertible), $f$ increases in some directions and decreases in others, and the Hessian is indefinite. A saddle can also have a zero Hessian, as for $f(x,y)=x^4-y^4$ at the origin.

Turn the surface: at a saddle point the tangent plane is horizontal, yet $f$ rises in some directions and falls in others.

Negating the definitions of a local maximum and minimum gives an equivalent reading: a stationary point $\boldsymbol{a}$ is a saddle point exactly when every neighbourhood of $\boldsymbol{a}$ contains points where $f$ is larger than $f(\boldsymbol{a})$ and points where it is smaller. For $x_1^2-x_2^2$ $x_{2}^{2}$ , moving a distance $t$ in the direction at angle $\theta$ to the $x_1$ -axis gives $f=t^2\cos2\theta$ $t^{2} cos 2 θ$ , positive near the $x_1$ -axis and negative near the $x_2$ -axis. The function $xy$ from the Examples has the same shape turned by $45^\circ$ , since $xy=\tfrac{1}{4}\big((x+y)^2-(x-y)^2\big)$ $y)^{2})$ .

An indefinite Hessian at a stationary point guarantees a saddle (unconstrained extrema conditions), but a saddle does not need one: $x_1^4-x_2^4$ has a saddle at the origin with zero Hessian. A semidefinite Hessian decides nothing: $x_1^2+x_2^4$ and $x_1^2-x_2^4$ both have Hessian $\operatorname{diag}(2,0)$ at the origin, the first has a minimum there and the second a saddle.

Key Results¶

An indefinite Hessian at a critical point is the standard second-order signal for a saddle point.
A zero gradient alone does not imply an extremum.
Some saddle points require higher-order information because the Hessian test can be inconclusive.

Examples¶

For $f(x,y)=x^2-y^2$ $y^{2}$ , $(0,0)$ is a saddle point because values increase along the $x$ -axis and decrease along the $y$ -axis.
For $f(x,y)=xy$ $x y$ , $(0,0)$ is a saddle point because $xy$ takes positive and negative values arbitrarily close to the origin.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 4, Lecture 1

Keywords: multivariable saddle Symbols: element of, nabla gradient, real numbers, subset, theta

Last updated: 20 October 2026 03:02