HomeAboutPurpose

About Critical Points Analyzer

What is the Critical Points Analyzer?

The Critical Points Analyzer is a web-based mathematical tool designed to help students, educators, and mathematics enthusiasts analyze and visualize critical points of functions of two variables. It identifies and classifies different types of critical points such as local maxima, local minima, and saddle points.

How it Works

When you enter a function expression, the analyzer:

  1. Finds critical points by solving where the gradient ∇f(x,y)=(0,0)\nabla f(x, y) = (0, 0)
  2. Calculates the Hessian matrix at each critical point
  3. Determines the nature of each critical point based on the eigenvalues of the Hessian
  4. Generates 2D and 3D visualizations of the function around critical points

Mathematical Background

Critical points of a function f(x,y)f(x, y) are points where both partial derivatives equal zero:

∂f∂x=0and∂f∂y=0\frac{\partial f}{\partial x} = 0 \quad \text{and} \quad \frac{\partial f}{\partial y} = 0

The classification of critical points is determined by analyzing the Hessian matrix:

H(f)=[∂2f∂x2∂2f∂x∂y∂2f∂y∂x∂2f∂y2]H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}

Local Maximum

All eigenvalues of the Hessian matrix are negative

Local Minimum

All eigenvalues of the Hessian matrix are positive

Saddle Point

Eigenvalues of the Hessian matrix have mixed signs

Features

LaTeX rendering of mathematical expressions
Interactive 2D and 3D visualizations
Support for a wide range of mathematical functions
Automatic calculation and classification of critical points
Example functions with varying critical point behaviors

References

Critical Points (Wikipedia)Hessian Matrix (Wikipedia)