Calculator Input
Use variables x and y.
Supported functions include sin, cos, tan, sqrt, abs, log, ln, exp, min, max, and pow.
Example Data Table
The following examples show valid formats and typical domains.
| Function | Domain | Use Case | Expected Behavior |
|---|---|---|---|
x^2 + y^2 - 4*x - 6*y + 13 |
-2 ≤ x ≤ 6, -1 ≤ y ≤ 7 | Convex surface | Minimum near (2, 3) |
sin(x) + cos(y) |
-3.14 ≤ x ≤ 3.14, -3.14 ≤ y ≤ 3.14 | Oscillating function | Several boundary and interior candidates |
x*y - x^2 - y^2 |
-4 ≤ x ≤ 4, -4 ≤ y ≤ 4 | Saddle style surface | Extrema often occur on boundary |
exp(-(x^2+y^2)) |
Disk centered at 0, radius 3 | Radial peak | Maximum near origin |
Formula Used
This calculator follows the closed and bounded domain idea from multivariable calculus. If a continuous function is checked on a closed bounded feasible region, absolute extrema can occur at interior critical points or along the boundary.
Numerical partials: fx ≈ [f(x+h,y) − f(x−h,y)] / 2h
Numerical partials: fy ≈ [f(x,y+h) − f(x,y−h)] / 2h
Hessian determinant: D = fxx fyy − (fxy)²
Absolute minimum: smallest accepted f(x,y) after boundary, interior, and refinement checks
Absolute maximum: largest accepted f(x,y) after boundary, interior, and refinement checks
How to Use This Calculator
- Enter a two variable function using
xandy. - Set the rectangular scan window with x and y limits.
- Choose an optional disk, ellipse, or linear half-plane constraint.
- Increase grid density for smoother plots and stronger estimates.
- Press calculate. The result appears above the form.
- Download the result table as CSV or PDF for study records.
Understanding Absolute Maximum and Minimum Values
What the calculator does
An absolute maximum is the largest value of a function on a chosen region. An absolute minimum is the smallest value. For a function of two variables, the surface may rise, fall, flatten, or bend in several directions. That makes visual checking useful, but it is not enough. A careful process must also test interior points and boundaries.
Why the domain matters
The same function can have different extrema on different regions. A wide rectangle may include a low valley. A disk may remove that valley. A linear restriction may cut away a peak. This is why the calculator asks for domain limits and optional constraints. The region defines which points are allowed.
Interior and boundary checks
Inside the region, the calculator looks for places where the gradient is small. These are possible critical points. On the boundary, the function behaves like a one dimensional path. Extrema can occur on edges, corners, circular arcs, elliptic arcs, or line boundaries. The calculator labels the likely location of each major candidate.
Numerical refinement
Many functions cannot be solved neatly by hand. This tool uses a grid search first. It then improves the best sampled points with local refinement. Higher grid density gives more starting points. More refinement passes can improve the final estimate. Very sharp peaks, discontinuities, and undefined expressions still need manual review.
Reading the graph
The graph shows the accepted surface values. Blank spaces are outside the feasible region or undefined. The minimum and maximum markers show the final estimates. Use the candidate table to compare f values, coordinates, locations, and gradient size. A small gradient suggests an interior critical point. A boundary label suggests the result depends on the region edge.
Best practice
Start with a broad domain and medium grid. Then narrow the domain around important candidates. Compare the result with calculus formulas. For exams, show the gradient equations, boundary testing, and candidate comparison table. For reports, export the data and include the plot with your explanation.
FAQs
1. What is an absolute maximum?
It is the greatest function value on the selected feasible region. It may occur inside the region, on a boundary, at a corner, or on a constraint curve.
2. What is an absolute minimum?
It is the smallest function value on the selected feasible region. The calculator compares sampled interior points, boundary points, and refined candidates.
3. Does this tool solve equations symbolically?
No. It gives numerical estimates. It uses grid sampling, finite differences, boundary checks, and local refinement to locate strong candidates.
4. Why do I need a domain?
Absolute extrema depend on the allowed region. Without a bounded domain, some functions may have no largest or smallest value.
5. What grid density should I use?
Use 41 to 81 for normal work. Use higher values for smoother graphs or complex functions, but expect slower calculations.
6. What does gradient magnitude mean?
It measures how close a point is to having zero first partial derivatives. Smaller values often indicate stronger interior critical candidates.
7. Can extrema occur on boundaries?
Yes. Boundary values are essential. Many multivariable functions reach absolute highs or lows on edges, corners, or constraint curves.
8. Why are my results approximate?
The calculator uses numerical methods. Increase grid density, refine passes, and compare results with manual derivative work for better confidence.