Saddle Point — Definition, Formula & Examples

A saddle point is a critical point of a function of two or more variables where the surface curves upward in one direction and downward in another, making it neither a local maximum nor a local minimum.

A point $(a, b)$ is a saddle point of $f(x, y)$ if $f_x(a,b) = 0$ and $f_y(a,b) = 0$ , but for every open disk centered at $(a, b)$ , there exist points where $f > f(a,b)$ and points where $f < f(a,b)$ . Equivalently, when the second-derivative test applies, the discriminant $D = f_{xx} f_{yy} - (f_{xy})^2$ is negative at that point.

Key Formula

D = f_{xx}(a,b)\, f_{yy}(a,b) - \bigl(f_{xy}(a,b)\bigr)^2

Where:

$D$ = Discriminant (also called the Hessian determinant) at the critical point
$f_{xx}$ = Second partial derivative of f with respect to x twice
$f_{yy}$ = Second partial derivative of f with respect to y twice
$f_{xy}$ = Mixed second partial derivative of f with respect to x then y

How It Works

To find saddle points, first locate all critical points by setting

f_x = 0

and

f_y = 0

simultaneously. Then apply the second-derivative test: compute

D = f_{xx} f_{yy} - (f_{xy})^2

at each critical point. If

D < 0

, the critical point is a saddle point. The name comes from the shape of a horse saddle, which curves up along the horse's spine and curves down on the sides.

Worked Example

Problem: Classify the critical point of f(x, y) = x² − y² at the origin.

Find critical points: Compute the first partial derivatives and set them to zero.

f_x = 2x = 0, \quad f_y = -2y = 0 \quad \Rightarrow \quad (x, y) = (0, 0)

Compute second partial derivatives: Find all three second-order partials needed for the discriminant.

f_{xx} = 2, \quad f_{yy} = -2, \quad f_{xy} = 0

Evaluate the discriminant: Plug into the formula for D.

D = (2)(-2) - (0)^2 = -4

Answer: Since D = −4 < 0, the origin is a saddle point. Along the x-axis the function curves upward (like x²), while along the y-axis it curves downward (like −y²).

Why It Matters

Saddle points appear throughout optimization, machine learning, and economics. In neural network training, the loss landscape is high-dimensional and saddle points vastly outnumber local minima, which directly affects how gradient descent algorithms behave. Recognizing saddle points also matters when using Lagrange multipliers to solve constrained optimization problems.

Common Mistakes

Mistake: Assuming every critical point that is not a local max must be a local min.

Correction: A critical point can be a saddle point — always check the sign of the discriminant D before classifying.