Understanding two variable extrema
A function of two variables describes a surface. Its high and low points often matter in calculus, economics, engineering, and optimization. This calculator helps inspect those points with a numerical method. It does not require symbolic differentiation. Instead, it estimates the first and second partial derivatives near many starting points.
Why critical points matter
Local extrema appear where the surface becomes flat in both directions. That means fx is near zero and fy is near zero. Such points are called stationary or critical points. A critical point can be a local minimum, local maximum, saddle point, or uncertain case. The final type appears when the second derivative test is too close to decide.
How the test works
The calculator builds a Hessian matrix from fxx, fyy, and fxy. Then it computes the determinant D. 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 near zero, more analysis is needed.
Numerical search notes
The search range creates starting guesses. Smaller seed spacing checks more places. It may find more points, but it also needs more time. The derivative step controls finite differences. Very small values can amplify rounding error. Very large values can hide sharp changes. Start with balanced values, then refine the report.
Practical use
Use clean formulas with x and y. Write powers with the caret symbol. Common functions include sin, cos, tan, exp, log, sqrt, and abs. Avoid discontinuities inside the search area. Review every result before using it in a proof or design decision. Numerical answers are best viewed as strong evidence. Exact calculus may still be needed for final confirmation.
Interpreting results
Check the gradient size first. Smaller values mean the point is closer to stationary. Then inspect D and fxx. Compare the function value with nearby sample values. Export the report when you need class notes, homework checks, or project documentation.
For difficult surfaces, repeat the run with wider ranges. Change one setting at a time. This keeps comparisons fair and makes errors easier to spot during later review.