Calculator Form
Formula Used
One variable: For f(x) = ax³ + bx² + cx + d, solve f'(x) = 3ax² + 2bx + c = 0. Then use f''(x) = 6ax + 2b.
Two variables: For f(x,y) = ax² + by² + cxy + dx + ey + k, solve fx = 2ax + cy + d = 0 and fy = 2by + cx + e = 0.
Hessian test: D = fxx fyy - fxy². If D > 0 and fxx > 0, it is a local minimum. If D > 0 and fxx < 0, it is a local maximum. If D < 0, it is a saddle point.
How to Use This Calculator
- Select single variable mode or two variable surface mode.
- Enter the matching coefficients in the form fields.
- Choose the number of decimal places for the result.
- Click the calculate button.
- Review the critical point, function value, test value, and classification.
- Use the CSV or PDF button to save the result.
Example Data Table
| Mode | Function | Expected Critical Point | Expected Type |
|---|---|---|---|
| Two Variable | x² + 2y² + xy - 6x - 8y | (1.6, 1.6) | Local minimum |
| Two Variable | -x² - y² + 4x + 2y | (2, 1) | Local maximum |
| Two Variable | x² - y² | (0, 0) | Saddle point |
| Single Variable | x³ - 3x² + 4 | x = 0, x = 2 | Maximum, minimum |
Understanding Critical Points
A local maximum is a nearby high point. A local minimum is a nearby low point. A saddle point is different. It rises in one direction. It falls in another direction. This calculator helps compare those cases with clear derivative tests.
For one variable curves, the first derivative gives critical x values. These values happen where the slope is zero. The second derivative then checks the bend. A positive second derivative means the graph bends upward. That point is a local minimum. A negative second derivative means the graph bends downward. That point is a local maximum. When the second derivative is zero, the test cannot decide alone.
For two variable quadratic surfaces, the calculator uses partial derivatives. It solves both gradient equations together. The gradient must be zero at a stationary point. Then the Hessian matrix gives the classification. The Hessian determinant is D = fxx fyy - fxy². If D is positive and fxx is positive, the point is a local minimum. If D is positive and fxx is negative, the point is a local maximum. If D is negative, the point is a saddle point. If D is zero, the result needs another method.
Why This Tool Helps
Manual work can be slow. Sign errors also happen often. This tool shows the formulas beside the result. It keeps inputs organized. It also supports exports for study notes, classroom checks, or project records. Use enough decimal places when coefficients are not whole numbers.
Good Use Cases
Use it for calculus practice. Use it for optimization examples. Use it for checking quadratic models. It can also support business, science, and engineering lessons where a peak, valley, or turning surface matters. The result should still be reviewed. The calculator follows standard derivative rules. It does not replace a full proof for complex functions. For higher degree or non-polynomial functions, use the result as a guide and continue with symbolic or graph based analysis.
Reading the Output
Check the critical point first. Then read the classification line. The derivative summary explains why that label appears. Save the CSV for spreadsheets. Save the PDF for printable records. Keep coefficients consistent. Small rounding choices can shift displayed values during final class reports.
FAQs
What is a local maximum?
A local maximum is a point where nearby function values are lower. It is not always the highest point on the full graph.
What is a local minimum?
A local minimum is a point where nearby function values are higher. It may be different from the absolute minimum.
What is a saddle point?
A saddle point is stationary, but it is not a local maximum or local minimum. The surface rises in one direction and falls in another.
What does the Hessian determinant show?
The Hessian determinant helps classify a two variable stationary point. It checks curvature using second partial derivatives.
Why is my result inconclusive?
The derivative test may fail when the second derivative or Hessian determinant is zero. More analysis is then needed.
Can this handle every function?
No. This page supports cubic single variable curves and quadratic two variable surfaces. Other functions need symbolic or numerical tools.
What does decimal precision control?
Decimal precision controls how many digits appear after the decimal point. It does not change the underlying calculation.
Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for printable notes or reports.