Calculator Inputs
This calculator studies the quadratic bivariate surface f(x,y) = ax² + bxy + cy² + dx + ey + f.
Example Data Table
| Function | Coefficients | Critical Point | Classification |
|---|---|---|---|
| f(x,y) = x² + y² - 4x - 6y + 13 | a=1, b=0, c=1, d=-4, e=-6, f=13 | (2, 3) | Local minimum |
| f(x,y) = -x² - y² + 8x + 2y | a=-1, b=0, c=-1, d=8, e=2, f=0 | (4, 1) | Local maximum |
| f(x,y) = x² - y² + 2x - 4y | a=1, b=0, c=-1, d=2, e=-4, f=0 | (-1, -2) | Saddle point |
Formula Used
Function: f(x,y) = ax² + bxy + cy² + dx + ey + f
Gradient: fₓ = 2ax + by + d, and fᵧ = bx + 2cy + e
Critical point condition: fₓ = 0 and fᵧ = 0
Hessian determinant: D = fₓₓfᵧᵧ − fₓᵧ²
If D > 0 and fₓₓ > 0, the point is a minimum. If D > 0 and fₓₓ < 0, the point is a maximum. If D < 0, the point is a saddle point.
How to Use This Calculator
- Enter the coefficients for the quadratic surface.
- Use positive or negative values as required.
- Set the x and y range when bounded checking is needed.
- Choose the number of decimal places for the output.
- Press calculate to show the result above the form.
- Use the CSV or PDF button to save your result.
Maxima, Minima, and Saddle Points Explained
What the Calculator Finds
Maxima, minima, and saddle points describe important locations on a surface. A local maximum is a point where nearby values are smaller. A local minimum is a point where nearby values are larger. A saddle point rises in one direction and falls in another. This calculator focuses on quadratic surfaces in two variables. These surfaces are common in algebra, optimization, economics, physics, and data fitting.
Why the Gradient Matters
The first step is finding where the gradient is zero. The gradient contains the partial derivatives with respect to x and y. When both derivatives are zero, the surface has no immediate uphill direction. That point is called a critical point. It may be a maximum, minimum, saddle point, or degenerate point. The gradient alone does not tell the full story.
Why the Hessian Test Helps
The Hessian matrix measures second-order curvature. It shows how the surface bends near the critical point. The determinant of this matrix is the key test value. A positive determinant means the surface bends the same way in both main directions. A negative determinant means the surface bends in opposite directions. That produces a saddle point.
Bounded Region Checking
Many real problems have limits. A design may require a fixed range. A budget model may allow only certain values. The bounded option checks corners, edges, and interior candidates. This helps locate global maximum and minimum values inside a rectangle. It is useful when unconstrained behavior is not enough.
Practical Use
Use the calculator to verify homework, build lesson examples, or review optimization cases. Change one coefficient at a time to see how the point type changes. The export buttons make it easier to keep records. The result table also supports reports and classroom notes.
FAQs
1. What type of function does this calculator support?
It supports quadratic functions in two variables using the form ax² + bxy + cy² + dx + ey + f.
2. What is a critical point?
A critical point occurs where both first partial derivatives equal zero. It is the main candidate for a maximum, minimum, or saddle point.
3. What does the Hessian determinant show?
It shows how the surface bends near the critical point. Its sign helps classify the point using the second derivative test.
4. When is a point a local minimum?
A point is a local minimum when the Hessian determinant is positive and fₓₓ is also positive.
5. When is a point a local maximum?
A point is a local maximum when the Hessian determinant is positive and fₓₓ is negative.
6. When is a point a saddle point?
A saddle point occurs when the Hessian determinant is negative. The surface rises in one direction and falls in another.
7. What does bounded mode do?
Bounded mode checks the selected rectangle. It compares corners, edge candidates, and interior points to find global values.
8. Why might the result be inconclusive?
The result may be inconclusive when the Hessian determinant is near zero. In that case, the usual second derivative test is weak.