Calculator Input
Use explicit multiplication, such as 3*x*y. Supported functions include
sin, cos, tan, exp, log,
sqrt, abs, and pow.
Plotly Graph
The contour plot shows the function surface level curves. Critical points appear as markers when results are available.
Example Data Table
| Function | Domain | Known critical points | Expected behavior |
|---|---|---|---|
x^2 + y^2 |
[-3, 3] × [-3, 3] | (0, 0) | Local minimum |
x^2 - y^2 |
[-3, 3] × [-3, 3] | (0, 0) | Saddle point |
-x^2 - y^2 + 4 |
[-3, 3] × [-3, 3] | (0, 0) | Local maximum |
x^3 - 3*x + y^3 - 3*y |
[-3, 3] × [-3, 3] | (-1,-1), (-1,1), (1,-1), (1,1) | Mixed extrema and saddle points |
Formula Used
For a two variable function, critical points occur where the gradient is zero: ∇f(x,y) = (fx, fy) = (0, 0).
This calculator estimates partial derivatives with central differences: fx ≈ [f(x+h,y) - f(x-h,y)] / 2h and fy ≈ [f(x,y+h) - f(x,y-h)] / 2h.
The Hessian matrix is H = [[fxx, fxy], [fxy, fyy]]. The determinant test uses D = fxx × fyy - fxy².
- If D > 0 and fxx > 0, the point is a local minimum.
- If D > 0 and fxx < 0, the point is a local maximum.
- If D < 0, the point is a saddle point.
- If D ≈ 0, the test is inconclusive.
How to Use This Calculator
- Enter a smooth function using variables
xandy. - Set the search interval for both variables.
- Choose grid seeds. More seeds can find more isolated points.
- Adjust tolerance if the function is flat or highly curved.
- Click the submit button to calculate results.
- Review the table, Hessian test, and contour graph.
- Use CSV or PDF export for reports and assignments.
Critical Points in Multivariable Calculus
Why Critical Points Matter
Critical points help explain the shape of a multivariable function. They show where a surface may rise, fall, flatten, or change direction. In two variables, the surface can form a bowl, hill, pass, ridge, or flat zone. These points are important in calculus, optimization, economics, physics, machine learning, and engineering design.
Gradient Meaning
The gradient points toward the fastest increase of a function. When both partial derivatives are zero, the surface has no immediate direction of increase. That location becomes a candidate for a minimum, maximum, or saddle point. The calculator searches for these candidates by solving the gradient equations numerically.
Hessian Meaning
The Hessian studies curvature near each candidate point. It checks how the surface bends in the x direction, y direction, and mixed direction. A positive bowl shape suggests a local minimum. A negative cap shape suggests a local maximum. A mixed bend suggests a saddle point.
Numerical Search
Many functions are difficult to solve by hand. This tool starts from many seed points across the selected domain. It then applies a Newton style search to move toward locations where the gradient is close to zero. Duplicate results are merged, so repeated discoveries appear once in the final table.
Accuracy Tips
Smooth functions work best. Use explicit multiplication. Choose a domain that contains the expected answer. Increase grid seeds when the function has many separate critical points. Decrease the finite difference step only when the function is stable. If the Hessian determinant is close to zero, the classification may need deeper analysis.
Practical Use
Use the graph with the table. Contour lines reveal local shape. Markers show where the calculator found stationary behavior. Export the result when you need a clean record. Always compare numerical answers with algebraic reasoning when exact proof is required.
FAQs
1. What is a critical point?
A critical point is a point where the gradient is zero or undefined. For smooth two variable functions, this usually means fx = 0 and fy = 0.
2. What variables does this calculator support?
This calculator supports functions written with x and y. It is designed for two variable multivariable calculus problems and Hessian classification.
3. Why do I need a search domain?
The domain tells the calculator where to look. A wider domain can find more points, but it may take longer and include unwanted solutions.
4. What does inconclusive mean?
Inconclusive means the Hessian determinant is near zero. The standard second derivative test cannot classify the point reliably.
5. Why should I use explicit multiplication?
Explicit multiplication avoids parser errors. Write 2*x instead of 2x, and write x*y instead of xy for clear function input.
6. Can this calculator prove exact answers?
No. It gives numerical estimates. For exact proof, solve the derivative equations symbolically and verify each point with calculus rules.
7. Why are some points missing?
Some points may be outside the domain, too flat, too sharp, or missed by the seed grid. Increase grid size or adjust tolerance.
8. What is a saddle point?
A saddle point rises in one direction and falls in another. It is not a local maximum or a local minimum.