Calculator Inputs
This version solves quadratic multivariable functions in two variables: f(x,y) = ax² + by² + cxy + dx + ey + f.
Plotly Graph
The 3D surface updates from the current input values. A marker appears when a unique stationary point exists.
Example Data Table
| a | b | c | d | e | f | Expected x* | Expected y* | Expected maximum value |
|---|---|---|---|---|---|---|---|---|
| -2 | -1.5 | 1 | 8 | 6 | 3 | 2.727273 | 2.909091 | 22.636364 |
Use the example row to test the calculator quickly. It produces a confirmed global maximum because the Hessian is negative definite.
Formula Used
For this calculator, the multivariable function is modeled as:
The stationary point comes from the first partial derivatives:
∂f/∂y = cx + 2by + e = 0
Solving the system gives:
y* = (cd - 2ae) / (4ab - c²)
The second derivative test uses the Hessian matrix:
[ c 2b ]
A unique maximum is confirmed when both conditions hold:
How to Use This Calculator
- Enter the six coefficients for the quadratic function.
- Choose the x and y viewing ranges for the graph.
- Set the grid points for plot smoothness.
- Press Find Maximum to compute the stationary point.
- Read the classification to confirm whether a maximum exists.
- Review the Hessian determinant and derivative values.
- Inspect the 3D Plotly surface for shape and peak location.
- Export the result as CSV or PDF when needed.
FAQs
1. What type of function does this calculator solve?
It solves quadratic functions in two variables. The model is ax² + by² + cxy + dx + ey + f. That makes the stationary point and maximum test exact, not approximate.
2. How does the calculator know a maximum exists?
It checks the Hessian matrix. A unique maximum exists when a is negative and 4ab − c² is positive. Those conditions make the surface curve downward in every local direction.
3. What happens when 4ab − c² equals zero?
The stationary system becomes singular or nearly singular. In that case, the calculator reports that the test is inconclusive or that no unique stationary point was found.
4. Can this page solve any multivariable expression?
This version focuses on two-variable quadratic functions. That keeps the results exact, stable, and easy to verify. More general functions usually need numerical optimization methods instead.
5. Why does the graph also show a window maximum?
The sampled window maximum helps you inspect the plotted region. It is useful when no analytical maximum is confirmed or when you want to compare the interior point against visible boundary behavior.
6. Does the constant term change the maximum location?
No. The constant term shifts the entire surface upward or downward. It changes the final function value, but it does not change the stationary coordinates.
7. What does a saddle point mean here?
A saddle point means the surface rises in one direction and falls in another. The stationary point exists, but it is neither a local maximum nor a local minimum.
8. Why should I increase grid points?
Higher grid points make the Plotly surface smoother and improve the sampled window estimate. Very high values also increase browser work, so moderate settings are usually best.