Critical Point Classification Calculator

Enter derivative data

Provide the gradient and second partial derivatives at one candidate point. The calculator applies the two-variable second derivative test and plots a local quadratic surface.

Function label

Display label for the analyzed function.

Point label

Short label for the candidate point.

Stationarity tolerance

Small threshold for near-zero derivatives.

x-coordinate at point

y-coordinate at point

Function value f(x₀, y₀)

First partial fₓ

First partial fᵧ

Second partial fₓₓ

Second partial fᵧᵧ

Mixed partial fₓᵧ

Plot range around point

Used for the local quadratic graph window.

Reset

Example data table

Case	Point	fₓ	fᵧ	fₓₓ	fᵧᵧ	fₓᵧ	D = fₓₓfᵧᵧ - fₓᵧ²	Classification
Quadratic bowl	(0, 0)	0	0	4	6	1	23	Local minimum
Upside-down bowl	(1, -1)	0	0	-5	-2	0.5	9.75	Local maximum
Hyperbolic shape	(2, 3)	0	0	2	-3	0	-6	Saddle point
Borderline test	(0, 2)	0	0	0	4	0	0	Inconclusive / Degenerate
Non-stationary	(1, 1)	0.3	-0.2	2	2	0	4	Not a critical point

Formula used

Gradient at the candidate point:
∇f(x₀, y₀) = (fₓ, fᵧ)

Hessian matrix:
H = [[fₓₓ, fₓᵧ], [fₓᵧ, fᵧᵧ]]

Second derivative test determinant:
D = fₓₓfᵧᵧ − (fₓᵧ)²

Hessian trace and eigenvalues:
tr(H) = fₓₓ + fᵧᵧ
λ₁,₂ = (tr(H) ± √((fₓₓ − fᵧᵧ)² + 4fₓᵧ²)) / 2

Classification rules:

If ||∇f|| is larger than tolerance, the point is not critical.
If both eigenvalues are positive, the point is a local minimum.
If both eigenvalues are negative, the point is a local maximum.
If eigenvalues have opposite signs, the point is a saddle point.
If any eigenvalue is near zero, the test is inconclusive and higher-order analysis may be needed.

Plot model: The graph uses the local second-order approximation z ≈ f₀ + fₓΔx + fᵧΔy + 0.5fₓₓΔx² + fₓᵧΔxΔy + 0.5fᵧᵧΔy².

How to use this calculator

Enter a function label and a point label for reporting.
Type the point coordinates and the function value at that point.
Provide first partial derivatives fₓ and fᵧ evaluated at the point.
Provide second partial derivatives fₓₓ, fᵧᵧ, and fₓᵧ at the same point.
Choose a tolerance for deciding whether derivatives are effectively zero.
Set a plot range to control the visible neighborhood around the point.
Press Classify Critical Point to show the result above the form.
Use the export buttons to save the computed summary as CSV or PDF.

FAQs

1) What does this calculator classify?

It classifies a candidate point for a two-variable function using the gradient and Hessian values you enter. It reports whether the point is a local minimum, local maximum, saddle point, degenerate case, or not critical.

2) Why do I need first derivatives?

A true critical point requires the gradient to be zero or very close to zero. Without that condition, the second derivative test should not be used to claim a local maximum, minimum, or saddle.

3) What does the determinant D tell me?

For two variables, D = fₓₓfᵧᵧ − fₓᵧ² summarizes Hessian curvature interaction. Positive D suggests same-sign curvature directions, while negative D indicates mixed curvature and therefore saddle behavior.

4) Why are Hessian eigenvalues shown?

Eigenvalues directly reveal local curvature directions. Both positive means upward opening, both negative means downward opening, and opposite signs mean the surface bends up in one direction and down in another.

5) What does inconclusive or degenerate mean?

It means the Hessian test is borderline, usually because the determinant or an eigenvalue is near zero. In that situation, higher-order terms may decide the actual local behavior.

6) What is the role of tolerance?

Tolerance treats tiny numerical values as practically zero. This is useful when derivatives come from measurement, estimation, or floating-point calculations rather than exact symbolic differentiation.

7) What does the Plotly graph represent?

The graph shows the local quadratic approximation around the chosen point, not the full original function. It is meant to visualize nearby shape and reinforce the classification result.

8) Can I use this for optimization studies?

Yes. It is useful for checking stationary candidates in multivariable optimization workflows, especially when you already know derivative values from symbolic work, software output, or numerical models.