Discriminant Calculator Multivariable

Multivariable Discriminant:

\[ D = \begin{vmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{vmatrix} \]

Discriminant Result:

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1. What is the Multivariable Discriminant?

The multivariable discriminant is the determinant of the Hessian matrix, which contains all second-order partial derivatives of a function. It's used to analyze critical points in multivariable calculus, determining whether they are local maxima, minima, or saddle points.

2. How Does the Calculator Work?

The calculator computes the determinant of the Hessian matrix:

\[ D = \begin{vmatrix} f_{x_1x_1} & f_{x_1x_2} & \cdots & f_{x_1x_n} \\ f_{x_2x_1} & f_{x_2x_2} & \cdots & f_{x_2x_n} \\ \vdots & \vdots & \ddots & \vdots \\ f_{x_nx_1} & f_{x_nx_2} & \cdots & f_{x_nx_n} \end{vmatrix} \]

Where:

\( f_{x_ix_j} \) — Second partial derivative with respect to x_i and x_j
\( n \) — Number of variables in the function

Explanation: The discriminant helps classify critical points found by setting the gradient to zero.

3. Importance of Discriminant Calculation

Details: The discriminant is crucial in optimization problems, economic modeling, and machine learning to understand the nature of critical points in multidimensional spaces.

4. Using the Calculator

Tips: Enter the number of variables (2-5) and your multivariable function. Use standard mathematical notation (x^2 for x², etc.).

5. Frequently Asked Questions (FAQ)

Q1: What does a positive discriminant indicate?
A: If D > 0 and f_xx > 0 at a critical point, it's a local minimum. If D > 0 and f_xx < 0, it's a local maximum.

Q2: What does a negative discriminant mean?
A: A negative discriminant indicates a saddle point at the critical point.

Q3: What if the discriminant is zero?
A: When D = 0, the test is inconclusive, and other methods must be used to classify the critical point.

Q4: How does this generalize the second derivative test?
A: For single-variable functions, this reduces to the standard second derivative test D = f''(x).

Q5: What are the limitations of this approach?
A: It only works for twice-differentiable functions and requires computing all second partial derivatives.