Calculator Input
Enter coefficients for f(x,y) = ax² + by² + cxy + dx + ey + g.
Formula Used
Quadratic function:
f(x,y) = ax² + by² + cxy + dx + ey + g
First partial derivatives:
fx = 2ax + cy + d
fy = 2by + cx + e
Critical point equations:
2ax + cy + d = 0
cx + 2by + e = 0
Hessian determinant:
D = fxxfyy - (fxy)² = 4ab - c²
Classification rules:
D > 0 and fxx > 0: local minimum.
D > 0 and fxx < 0: local maximum.
D < 0: saddle point.
D = 0: degenerate or inconclusive case.
How to Use This Calculator
- Write your function in the form ax² + by² + cxy + dx + ey + g.
- Enter all six coefficients into the calculator fields.
- Set graph limits for x and y.
- Add a test point if you want a gradient check.
- Press Calculate to view the local extrema result.
- Use CSV for spreadsheet records.
- Use PDF for a printable report.
Example Data Table
| Function | Coefficients | Expected Result | Critical Set |
|---|---|---|---|
| x² + y² - 4x - 6y + 13 | a=1, b=1, c=0, d=-4, e=-6, g=13 | Local minimum | (2, 3) |
| -x² - y² + 4x + 6y | a=-1, b=-1, c=0, d=4, e=6, g=0 | Local maximum | (2, 3) |
| x² - y² | a=1, b=-1, c=0, d=0, e=0, g=0 | Saddle point | (0, 0) |
| x² + 2xy + y² | a=1, b=1, c=2, d=0, e=0, g=0 | Non-isolated minimum | x + y = 0 |
Local Extrema in Two Variables
A local extrema problem studies the shape of a surface near a point. For a function f(x, y), a local minimum is a low point in a small neighborhood. A local maximum is a high point in a small neighborhood. A saddle point rises in one direction and falls in another direction.
This calculator focuses on quadratic surfaces. It uses the form ax² + by² + cxy + dx + ey + g. This form is common in multivariable calculus, optimization, economics, engineering, and data fitting. The tool finds the critical point by solving the gradient equations. Then it uses the Hessian determinant to classify the point.
Why the Hessian Test Matters
The gradient tells where the surface becomes flat. A flat tangent plane is necessary for a local extremum. It is not enough. The Hessian matrix measures second derivative curvature. Its determinant shows whether the surface bends the same way in both main directions or bends in opposite ways.
When the determinant is positive and fxx is positive, the point is a local minimum. When the determinant is positive and fxx is negative, the point is a local maximum. When the determinant is negative, the point is a saddle point. When the determinant is zero, the standard test is inconclusive or degenerate.
Practical Uses
Local extrema appear in profit models, cost models, heat surfaces, response surfaces, and least squares problems. A business may maximize profit using two decision variables. An engineer may minimize error between a model and measured data. A scientist may study where a surface is stable or unstable.
Reading the Output
The result section shows the function, gradient, Hessian matrix, determinant, critical point, classification, and function value. The plot gives a quick visual check. The CSV export is useful for records. The PDF button helps save a formatted report.
Good Input Tips
Use real numeric coefficients. Keep graph limits reasonable. Wider ranges can hide local behavior. Smaller ranges near the critical point show curvature better. If the determinant is zero, read the note carefully. The surface may have a line of critical points, no stationary point, or a flat constant behavior. Always review behavior carefully.
FAQs
1. What is a local extrema point?
A local extrema point is a nearby high or low point on a surface. It can be a local maximum or local minimum.
2. What function type does this calculator support?
It supports two variable quadratic functions written as ax² + by² + cxy + dx + ey + g.
3. What is a critical point?
A critical point occurs where both first partial derivatives equal zero, or where the gradient becomes the zero vector.
4. What does the Hessian determinant show?
It shows the second derivative curvature pattern. It helps decide whether a critical point is a minimum, maximum, or saddle point.
5. What means D is less than zero?
D less than zero means the surface bends upward in one direction and downward in another. The point is a saddle point.
6. What happens when D equals zero?
The usual second derivative test is degenerate. The calculator checks for flat lines, constant surfaces, or missing stationary points.
7. Can I export the result?
Yes. Use the CSV button for spreadsheet output. Use the PDF button after calculation for a formatted report.
8. Why should I adjust graph limits?
Graph limits control the visible surface area. Smaller limits near the critical point often show local curvature more clearly.