Critical Point Calculator for f(x,y)

Calculator Input

Function f(x,y)

Use *, /, ^, sin(), cos(), exp(), log(), sqrt(), pi, and e.

Search mode

Scan steps per axis

Starting x

Starting y

Decimal places

x minimum

x maximum

y minimum

y maximum

Derivative step

Gradient tolerance

Maximum iterations

Example Data Table

These examples show how different surfaces can be tested.

Function	Suggested range	Expected behavior
`x^2 + y^2 - 4x - 6y + 13`	-5 to 5	One local minimum near (2, 3).
`x^2 - y^2`	-3 to 3	One saddle point near (0, 0).
`-x^2 - y^2 + 8`	-4 to 4	One local maximum near (0, 0).
`x^4 + y^4 - 4x^2 - 2y^2`	-3 to 3	Several stationary points may appear.

Formula Used

Gradient condition: Critical points occur where fx = ∂f/∂x = 0 and fy = ∂f/∂y = 0.

Hessian matrix: H = [[fxx, fxy], [fxy, fyy]].

Second derivative test: D = fxx*fyy - fxy^2.

Classification: If D > 0 and fxx > 0, the point is a local minimum. If D > 0 and fxx < 0, it is a local maximum. If D < 0, it is a saddle point.

The calculator estimates derivatives by central differences. Newton iteration then solves the two gradient equations. Grid mode repeats the solve from many starting points and removes duplicate solutions.

How to Use This Calculator

Enter a function using variables x and y.
Select grid scan when you want several possible critical points.
Set the x and y ranges around the region you want to study.
Use a smaller tolerance for stricter convergence.
Press calculate and review the result table above the form.
Use CSV for spreadsheet work or PDF for a printable summary.

Critical Point Calculator Guide

What This Tool Does

A critical point of a two variable function is a place where the surface stops rising or falling in every first order direction. The gradient becomes zero there. This calculator studies functions written as f(x,y). It estimates first derivatives, solves the gradient equations, and then reviews the second derivative test.

Why Critical Points Matter

Critical points are useful in optimization, engineering, economics, physics, data science, and geometry. A local minimum may show the least cost. A local maximum may show the best output. A saddle point may look flat, yet the surface rises in one direction and falls in another.

How the Search Works

The calculator uses a numerical method. It samples derivatives with central differences. It then applies Newton iteration to move toward a point where fx and fy are close to zero. Grid scan mode starts the process from many seed points. This helps find more than one solution inside the selected range. Duplicate points are merged, so the final table stays readable.

Reading the Result Table

Each row gives x, y, f(x,y), first derivatives, second derivatives, Hessian determinant, and classification. A small residual means the gradient is close to zero. The Hessian determinant decides the usual local shape. Positive D with positive fxx suggests a valley. Positive D with negative fxx suggests a peak. Negative D suggests a saddle.

Practical Tips

Use clear multiplication signs, such as 3*x*y. Choose a range that covers the visible region of interest. Increase scan steps when a function has several turning points. Lower the derivative step only when the function is smooth. Some flat or non-smooth surfaces may return inconclusive results. In those cases, compare the graph, table values, and nearby function behavior.

Limits to Remember

Numerical solving depends on the starting range and the shape of the function. Discontinuous formulas, sharp corners, hidden asymptotes, and very large powers can make estimates unstable. Treat the output as a guide, not a formal proof. For homework or research, verify important points by differentiating symbolically when possible. Then test nearby values. The plot can also reveal missed branches. Repeat scans after changing bounds.

FAQs

1. What is a critical point of f(x,y)?

It is a point where both first partial derivatives are zero or undefined. This calculator searches for points where fx and fy are close to zero.

2. Can this find every critical point?

No numerical scan can guarantee every point. A wider range, more scan steps, and good starting values improve the chance of finding more solutions.

3. Why does the result say inconclusive?

The second derivative test fails when the Hessian determinant is near zero. The point may be flat, degenerate, or need another analysis method.

4. Which expression syntax should I use?

Use x and y as variables. Write multiplication with *. Powers use ^. Supported functions include sin, cos, tan, exp, log, sqrt, and abs.

5. What does the residual mean?

The residual is the gradient size at the reported point. Smaller values mean fx and fy are closer to zero.

6. Why should I use grid scan?

Grid scan starts Newton solving from many seed points. This is useful when a function has multiple minima, maxima, or saddle points.

7. What derivative step is best?

The default step works for many smooth functions. Try smaller values for smooth curves and larger values when numerical noise appears.

8. Can I export the results?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report with the main table.