Calculator Input
This tool analyzes the quadratic form
f(x,y) = ax² + by² + cxy + dx + ey + k.
Example Data Table
| Function | a | b | c | d | e | k | Expected Result |
|---|---|---|---|---|---|---|---|
| 2x² + 3y² - 4xy - 8x - 2y + 5 | 2 | 3 | -4 | -8 | -2 | 5 | Minimum at (7, 5), value -28 |
| x² - y² + 4x - 2y + 1 | 1 | -1 | 0 | 4 | -2 | 1 | Saddle point |
| -3x² - 2y² + xy + 6x - 4y | -3 | -2 | 1 | 6 | -4 | 0 | Local maximum |
Formula Used
The calculator uses the general quadratic surface:
f(x,y) = ax² + by² + cxy + dx + ey + k
First partial derivatives are:
fₓ = 2ax + cy + d and
fᵧ = cx + 2by + e.
A stationary point is found by solving:
2ax + cy + d = 0 and
cx + 2by + e = 0.
The Hessian matrix is:
H = [[2a, c], [c, 2b]].
Its determinant is:
D = 4ab - c².
- If
D > 0andfₓₓ > 0, the point is a local minimum. - If
D > 0andfₓₓ < 0, the point is a local maximum. - If
D < 0, the point is a saddle point. - If
D = 0, the test is inconclusive.
How to Use This Calculator
- Enter the coefficients for the quadratic function.
- Use positive, negative, decimal, or zero values.
- Select decimal precision for cleaner results.
- Enable domain bounds when x and y have limits.
- Press the calculate button.
- Review the stationary point and Hessian classification.
- Check boundary candidates if a domain is enabled.
- Download the result as CSV or PDF for records.
Understanding Extrema of Two Variable Functions
Understanding Extremum of f(x,y)
An extremum is a highest or lowest value near a point. For a two variable function, the graph is a surface. The surface may rise, fall, or bend in different directions. This calculator focuses on quadratic surfaces. That choice keeps the method clear, fast, and useful for many models.
Why the Gradient Matters
A local maximum or minimum often appears where both first partial derivatives are zero. These equations form the gradient test. When the gradient is zero, the point is called stationary. A stationary point is not always an extremum. It can also be a saddle point. A saddle rises in one direction and falls in another direction.
Using the Hessian Test
The Hessian matrix stores second derivative information. Its determinant tells how the surface bends. A positive determinant with positive fxx suggests a local minimum. A positive determinant with negative fxx suggests a local maximum. A negative determinant suggests a saddle point. A zero determinant means the test cannot decide.
Boundary and Practical Checks
Many real problems have allowed ranges. For example, x and y may have lower and upper limits. Then the best value may occur on an edge or at a corner. This tool can compare the stationary point with boundary candidates. It also checks corners, because corners often control limited domains.
Interpreting Results
The result gives the critical point, function value, Hessian determinant, and classification. Use enough decimal precision to match your data. If the Hessian is degenerate, review the original model. If bounds are active, compare the interior result with the domain minimum and maximum. This gives a safer decision.
Good Modeling Practice
Always confirm that coefficients match your real function. Units should stay consistent. A mathematical extremum may not be practical if it lies outside allowed values. Boundary review helps avoid that error. Export the table when you need a record. Use the formula section to explain each step in reports.
Study Tip
For a better workflow, test several coefficient sets. Save each output. Compare signs, magnitudes, and domain limits. This habit reveals sensitivity. It also helps students see how small changes can move or remove an extremum completely.
FAQs
1. What does extremum mean?
An extremum is a maximum or minimum value. For f(x,y), it is a point where the surface reaches a high or low value locally or within a chosen domain.
2. What is a stationary point?
A stationary point occurs where both first partial derivatives equal zero. It may be a minimum, maximum, saddle point, or an inconclusive case.
3. What does the Hessian determinant show?
The Hessian determinant shows how the surface bends near a stationary point. Its sign helps classify the point using the second derivative test.
4. What is a saddle point?
A saddle point is not a maximum or minimum. The function rises in one direction and falls in another direction near the same point.
5. Why use boundary checks?
Boundary checks matter when x and y have limits. A function may reach its domain maximum or minimum on an edge or corner.
6. Can this calculator handle all functions?
This calculator is designed for quadratic two variable functions. General symbolic functions need a broader algebra system or numerical optimization method.
7. What happens if the determinant is zero?
If the determinant is zero, the standard Hessian test cannot decide. The function may need extra analysis, factoring, graphing, or directional testing.
8. What should I export?
Export the result table when you need records for homework, reports, quality checks, or repeated comparisons between different coefficient sets.