Calculator Input
This calculator evaluates quadratic two variable functions: f(x,y) = ax² + by² + cxy + dx + ey + k.
Example Data Table
Use these sample coefficients to test the calculator quickly.
| Example | a | b | c | d | e | k | Expected Behavior |
|---|---|---|---|---|---|---|---|
| Positive bowl | 1 | 2 | 1 | -6 | -8 | 12 | Local minimum |
| Negative cap | -2 | -1 | 0.5 | 4 | 3 | 5 | Local maximum |
| Saddle form | 1 | -1 | 0 | 0 | 0 | 0 | Saddle point |
Formula Used
Function: f(x,y) = ax² + by² + cxy + dx + ey + k
First partial derivative: fx = 2ax + cy + d
Second partial derivative: fy = 2by + cx + e
Critical point equations: 2ax + cy + d = 0 and cx + 2by + e = 0
Hessian matrix: H = [[2a, c], [c, 2b]]
Hessian determinant: D = (2a)(2b) - c² = 4ab - c²
Test: D > 0 and 2a > 0 means minimum. D > 0 and 2a < 0 means maximum. D < 0 means saddle.
How to Use This Calculator
- Enter the six coefficients for your quadratic function.
- Use zero for any missing term.
- Turn on boundary analysis when a rectangular domain is required.
- Enter minimum and maximum values for x and y.
- Select decimal precision for cleaner answers.
- Press the calculate button.
- Review the critical point, Hessian test, and boundary table.
- Download the result as a CSV or PDF report.
Understanding Maxima and Minima of Two Variables
Why Two Variable Optimization Matters
Many real problems depend on two changing inputs. A business may compare price and demand. An engineer may compare length and width. A student may study a surface in space. Maxima and minima help locate the highest and lowest useful values. They also show how a surface behaves near important points.
Critical Points
A critical point appears when both first partial derivatives equal zero. At that location, the surface has no immediate upward or downward slope in the x or y direction. This point may be a minimum, a maximum, or a saddle. The calculator solves the two derivative equations for quadratic functions. It then evaluates the original function at the critical point.
The Hessian Test
The Hessian matrix uses second partial derivatives. It measures the bending of the surface. A positive Hessian determinant with positive fxx usually means a local minimum. A positive determinant with negative fxx usually means a local maximum. A negative determinant means the surface bends upward in one direction and downward in another. That case is called a saddle point.
Boundary Checks
A local result is not always the final answer on a limited domain. When bounds are used, the calculator also checks corners and edge candidates. This is important for rectangles such as x between two values and y between two values. The smallest and largest candidate values become the bounded minimum and maximum.
Practical Use
Use this tool for calculus homework, modeling tasks, economics, geometry, and surface analysis. It gives derivative formulas, classification, candidate points, and export options. The grid scan is only an estimate. The analytic candidate table should be used for final bounded quadratic answers.
FAQs
1. What type of function does this calculator support?
It supports quadratic functions of two variables in the form ax² + by² + cxy + dx + ey + k.
2. What is a critical point?
A critical point is where both first partial derivatives equal zero. It may be a minimum, maximum, saddle point, or inconclusive case.
3. What does the Hessian determinant mean?
The Hessian determinant helps classify the surface near a critical point. It compares second derivative curvature in both variable directions.
4. When should I use boundary analysis?
Use boundary analysis when x and y are restricted to a rectangle. The highest or lowest value may occur on an edge or corner.
5. What does a saddle point mean?
A saddle point rises in one direction and falls in another. It is not a local maximum or local minimum.
6. Can this find global extrema?
For bounded rectangular domains, it compares interior, edge, and corner candidates. For unbounded quadratics, global behavior depends on curvature.
7. Why is the grid scan included?
The grid scan gives a quick numerical estimate across the selected rectangle. It supports checking, but analytic candidates remain more exact.
8. What should I enter for missing terms?
Enter zero for any missing coefficient. For example, if there is no xy term, set c equal to zero.