Calculator Form
Example Data Table
| Function | Point | Expected Type | Search Range |
|---|---|---|---|
| x^2 + y^2 | (0, 0) | Local minimum | -5 to 5 |
| -x^2 - y^2 | (0, 0) | Local maximum | -5 to 5 |
| x^2 - y^2 | (0, 0) | Saddle point | -5 to 5 |
| x^2 + y^2 + 2*x - 4*y + 5 | (-1, 2) | Local minimum | -5 to 5 |
Formula Used
The calculator estimates first partial derivatives with central differences. It uses fx = [f(x+h,y)-f(x-h,y)] / 2h. It also uses fy = [f(x,y+h)-f(x,y-h)] / 2h.
The Hessian matrix is built from fxx, fyy, and fxy. The determinant is D = fxx*fyy - fxy^2. 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, the test may be inconclusive.
How to Use This Calculator
Enter a function using x and y as variables. Choose the point where you want to test behavior. Set a small derivative step for numerical differentiation. Use a tolerance that matches your accuracy needs. Add bounded grid limits to search for approximate extremes. Press calculate. The result appears above the form and below the header. Download the result table when you need a saved copy.
About Multivariable Maxima and Minima
What the Tool Measures
A multivariable function can rise, fall, bend, and flatten in many directions. This makes its maximum and minimum behavior richer than a one variable curve. The calculator checks a selected point by estimating partial derivatives. It also studies the second derivative structure through the Hessian matrix. These values help decide whether the point behaves like a hill, valley, saddle, or uncertain surface.
Why Gradients Matter
The gradient contains the first partial derivatives. At a true interior critical point, both first partial derivatives are zero. Numerical work rarely gives exact zero. That is why this page includes tolerance. A small gradient magnitude means the function is nearly flat at the selected point. A large value means the point is probably not stationary.
Why the Hessian Matters
The Hessian describes local curvature. Positive curvature in every direction suggests a local minimum. Negative curvature in every direction suggests a local maximum. Mixed curvature suggests a saddle point. The determinant test summarizes this behavior for two variables. Eigenvalues add another useful view of the same curvature pattern.
Bounded Range Search
Local tests only describe behavior near one point. A function can have different behavior elsewhere. The grid search scans a rectangular region. It reports approximate smallest and largest sampled values. This does not replace symbolic optimization. It gives a practical check when exact solving is difficult. More grid points improve coverage. They also increase calculation time.
Good Input Practices
Use clear multiplication signs, such as 2*x*y. Avoid writing 2xy. Choose ranges that include the region of interest. Start with a moderate grid size. Then refine the search near promising points. Use smaller derivative steps for smooth functions. Use larger steps when the function has numerical noise. Always review results with mathematical judgment. Calculators support study, but context still matters.
FAQs
What is a critical point?
A critical point is a point where first partial derivatives are zero or undefined. This calculator checks near-zero gradient values using your tolerance setting.
What does the Hessian determinant show?
It shows local curvature behavior. A positive determinant can indicate a maximum or minimum. A negative determinant indicates a saddle point.
Can this solve symbolic derivatives?
No. It uses numerical differentiation. It estimates derivatives from nearby function values, then applies standard second derivative tests.
Why does the result say inconclusive?
The Hessian determinant may be near zero. In that case, the standard second derivative test cannot classify the point reliably.
What grid size should I use?
Use 21 to 51 points for quick checks. Use higher values for smoother scans, but expect more calculation time.
Does grid search find exact extrema?
No. It samples points inside your chosen range. It gives an approximation, not a guaranteed exact global result.
Which functions are supported?
The calculator supports common functions like sin, cos, tan, sqrt, log, exp, abs, pow, min, and max.
Why should I use multiplication signs?
The evaluator needs explicit multiplication. Write 3*x*y instead of 3xy. This prevents parsing errors and improves accuracy.