Calculator Inputs
Example Data Table
| Function | Hessian Determinant | Critical Point | Classification |
|---|---|---|---|
| f = x² + 2y² + xy - 4x - 6y + 3 | 7 | (10/7, 8/7) | Local minimum |
| f = -2x² - y² + 4x + 2y + 8 | 16 | (1, 1) | Local maximum |
| f = x² - y² + 3x - 2y | -4 | (-3/2, -1) | Saddle point |
| f = x² + 2xy + y² | 0 | Not unique | Inconclusive |
Formula Used
The calculator uses the quadratic model:
f(x,y) = ax² + by² + cxy + dx + ey + k
The first partial derivatives are:
fx = 2ax + cy + d
fy = cx + 2by + e
A stationary point is found by solving:
fx = 0 and fy = 0
The Hessian matrix is:
H = [[2a, c], [c, 2b]]
The determinant is:
D = fxxfyy - fxy² = 4ab - c²
When D is positive and fxx is positive, the point is a local minimum. When D is positive and fxx is negative, the point is a local maximum. When D is negative, the point is a saddle point. When D is zero, the test is inconclusive.
How to Use This Calculator
- Enter the coefficient of x² in the a field.
- Enter the coefficient of y² in the b field.
- Enter the coefficient of xy in the c field.
- Enter the linear coefficients d and e.
- Enter the constant term k.
- Set decimal places for the displayed answer.
- Enable the bounded region option when you need a rectangle check.
- Press Calculate to view the result below the header.
- Use CSV or PDF buttons to download the same report.
Extrema Study Guide
Why Multivariable Extrema Matter
Multivariable extrema appear when a surface stops rising or falling in every chosen direction. This calculator focuses on quadratic two variable models because they are common in algebra, calculus, physics, and optimization lessons. A quadratic surface gives exact derivatives, a clean Hessian matrix, and a reliable second derivative test.
What the Tool Calculates
The tool accepts coefficients for x squared, y squared, the mixed xy term, both linear terms, and the constant term. It then forms the gradient equations. A stationary point is found where both partial derivatives equal zero. When the determinant of the Hessian is not zero, the point is unique. If the determinant is zero, the model may have a flat direction, a line of stationary points, or an inconclusive second derivative test.
How Classification Works
Classification uses the determinant and the first second partial derivative. A positive determinant with positive fxx means a local minimum. A positive determinant with negative fxx means a local maximum. A negative determinant means a saddle point. A zero determinant needs deeper checking because nearby values may behave in a delicate way.
Why the Hessian Helps
This method is powerful because it connects symbols with geometry. The Hessian describes local curvature. Eigenvalue signs show whether the surface bends upward, downward, or in opposite directions. The calculated value f(x,y) helps compare the critical point with optional reference points or domain limits.
Best Use Cases
Students can use the example table to test known surfaces before entering homework data. Teachers can export reports for worksheets. Analysts can model simple cost, profit, energy, or error surfaces. The calculator is not meant to replace full symbolic algebra for every possible function. It gives a dependable workflow for quadratic models and teaches the main extrema test clearly.
Important Checking Advice
For bounded regions, also check boundary curves and corner points. A local minimum inside the region may not be the absolute minimum. A saddle point may still appear between useful feasible points. Always read the classification, the Hessian determinant, and the function value together. That combined view gives a stronger interpretation than a single number.
Practical Notes
Use sensible units for every coefficient. Large numbers can make reports harder to read. Scale variables when needed. Keep notes beside each exported file, so later revisions remain easy to audit and explain clearly with confidence.
FAQs
1. What type of function does this calculator support?
It supports two variable quadratic functions in the form ax² + by² + cxy + dx + ey + k. This form is ideal for Hessian based extrema tests.
2. What is a stationary point?
A stationary point is where both first partial derivatives equal zero. At that point, the surface has no immediate increase or decrease in the x or y direction.
3. How does the calculator classify extrema?
It uses the Hessian determinant and fxx. Positive determinant with positive fxx gives a minimum. Positive determinant with negative fxx gives a maximum. Negative determinant gives a saddle point.
4. What does an inconclusive result mean?
It means the Hessian determinant is zero. The usual second derivative test cannot decide the behavior. More algebra, graphing, or directional checking may be needed.
5. Can this find absolute extrema?
Yes, when you enable bounded region checking. It tests the interior point, corners, and valid edge candidates inside the rectangle you enter.
6. Why are eigenvalues shown?
Eigenvalues describe curvature directions. Positive eigenvalues suggest upward bending. Negative eigenvalues suggest downward bending. Mixed signs indicate saddle behavior.
7. What should I enter for a missing term?
Enter zero for any missing coefficient. For example, if there is no xy term, place 0 in the c field.
8. Can I download my calculation?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable report with the function, derivatives, Hessian, and result.