Discriminant Formula:
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The discriminant (D) is used in multivariable calculus to classify critical points of functions of two variables. It is calculated from the second partial derivatives of the function and helps determine whether a critical point is a local maximum, local minimum, or saddle point.
The calculator uses the discriminant formula:
Where:
Interpretation:
Details: The discriminant is crucial for analyzing the behavior of multivariable functions at critical points, which has applications in optimization problems across physics, engineering, and economics.
Tips: Enter the values of the second partial derivatives \( f_{xx} \), \( f_{yy} \), and the mixed partial derivative \( f_{xy} \). The calculator will compute the discriminant and you can interpret the result using the rules above.
Q1: What if the discriminant equals zero?
A: When D = 0, the second derivative test is inconclusive. You may need to use other methods to analyze the critical point.
Q2: Can this be used for functions with more than two variables?
A: No, this discriminant is specifically for functions of two variables. Higher dimensions require more complex analysis.
Q3: How do I find the partial derivatives?
A: Partial derivatives are calculated by differentiating the function with respect to one variable while treating others as constants.
Q4: What's the difference between fxy and fyx?
A: For continuous functions with continuous partial derivatives (as typically encountered), fxy = fyx by Clairaut's theorem.
Q5: Can this calculator handle complex numbers?
A: No, this calculator is designed for real-valued functions of real variables.