Calculator Inputs
Use the quadratic form f(x,y,z)= ax² + by² + cz² + dxy + exz + fyz + gx + hy + iz + j.
Formula Used
The calculator uses this quadratic function:
f(x,y,z)= ax² + by² + cz² + dxy + exz + fyz + gx + hy + iz + j
The gradient is:
fx = 2ax + dy + ez + g
fy = 2by + dx + fz + h
fz = 2cz + ex + fy + i
The stationary point solves ∇f = 0. The Hessian matrix is:
[[2a, d, e], [d, 2b, f], [e, f, 2c]]
Positive definite means local minimum. Negative definite means local maximum. Mixed curvature means saddle point.
How to Use This Calculator
- Enter coefficients for squared, mixed, linear, and constant terms.
- Use zero for any term that is not present.
- Enter a checking point to evaluate a chosen location.
- Choose decimal precision for cleaner output.
- Press the calculate button to see results above the form.
- Download the CSV or PDF file after calculation.
Example Data Table
| Function | Key coefficients | Stationary point | Expected result |
|---|---|---|---|
| x² + y² + z² - 4x - 6y - 8z | a=1, b=1, c=1, g=-4, h=-6, i=-8 | (2, 3, 4) | Local minimum |
| -x² - y² - z² + 2x + 4y + 6z | a=-1, b=-1, c=-1, g=2, h=4, i=6 | (1, 2, 3) | Local maximum |
| x² - y² + z² | a=1, b=-1, c=1 | (0, 0, 0) | Saddle point |
Extrema of Three Variable Functions
Why Extrema Matter
Three variable extrema matter when a value depends on x, y, and z. Many models use this pattern. A cost surface can depend on length, width, and depth. A physics model can depend on position in space. A data model can depend on three controllable inputs. This calculator focuses on a quadratic form because it is common, readable, and useful for exact Hessian testing.
How the Tool Works
The tool writes the function as a sum of squared terms, mixed terms, linear terms, and a constant. It then builds the gradient. The gradient contains three first partial derivatives. A stationary point appears where all three derivatives become zero. For a quadratic function, this system is linear. The calculator solves that system with determinants. It also reports the Hessian matrix, so the reasoning stays visible.
Reading the Classification
Classification comes from the Hessian. A positive definite Hessian gives a local minimum. A negative definite Hessian gives a local maximum. An indefinite Hessian gives a saddle point. A singular Hessian may mean the test is not enough. In that case, more analysis is needed. This is why the result includes determinants and principal minors, not only a final label.
Checking Points
The calculator also accepts a checking point. You can enter any x, y, and z values. The tool evaluates the function and gradient at that point. This helps compare a guessed point with the computed stationary point. It also supports lessons, homework review, and model checking.
Input Tips
Use clean coefficients for the best result. Put zero in unused terms. Mixed terms can change the classification strongly. Linear terms move the stationary point. Constant terms change the function value, but they do not move the point. The example table shows practical combinations and their outcomes.
Saving Results
Exports help save the work. The CSV file is useful for spreadsheets. The PDF file is useful for quick records. Both include the main inputs and classification. You can keep them with project notes or class solutions.
Study Note
This calculator is an educational helper. It explains the derivative test clearly. It does not replace full proof work for complex nonquadratic functions. Still, it gives a strong starting point for three variable extrema problems.
Always review domain limits and units. Check modeling assumptions before final decisions in practice today.
FAQs
1. What does this calculator find?
It finds the stationary point of a three variable quadratic function. It also classifies the point as a local minimum, local maximum, saddle point, or inconclusive case.
2. What function format should I use?
Use the displayed quadratic format. Enter each coefficient separately. Put zero in a field when that term is missing from your function.
3. Can it solve nonquadratic functions?
This version is designed for quadratic functions. Nonquadratic functions may require symbolic differentiation, numerical solvers, or additional tests beyond this calculator.
4. What is the Hessian matrix?
The Hessian matrix contains second partial derivatives. It describes curvature near a stationary point and helps classify the local behavior of the function.
5. What does saddle point mean?
A saddle point rises in some directions and falls in others. It is stationary, but it is not a local maximum or minimum.
6. Why is my result inconclusive?
The Hessian determinant may be zero. That means the second derivative test cannot make a final classification. More detailed analysis may be needed.
7. What is the check point for?
The check point lets you evaluate the function and gradient at any chosen x, y, and z values. It helps verify guesses and compare points.
8. What exports are available?
You can download the result as CSV or PDF. CSV works well for spreadsheets. PDF works well for saved notes and printable reports.