Find ((∂^2 f)/(∂x^2))((∂^2 f)/(∂y^2))-((∂^2 f)/(∂x ∂y))^2 for the following functions: a f(x, y)

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Find ((∂^2 f)/(∂x^2))((∂^2 f)/(∂y^2))-((∂^2 f)/(∂x ∂y))^2...

Find $\left(\frac{\partial^{2} f}{\partial x^{2}}\right)\left(\frac{\partial^{2} f}{\partial y^{2}}\right)-\left(\frac{\partial^{2} f}{\partial x \partial y}\right)^{2}$ for the following functions:
a $f(x, y)=x^{3}+y^{3}-3 x y$
b $f(x, y)=2 x y+\frac{2000}{x}+\frac{2000}{y}$

Key Concepts

Partial Derivatives

Partial derivatives measure the rate at which a function changes as only one of its independent variables is varied, keeping the other variables constant. They are fundamental in analyzing multivariable functions and form the basis for more advanced concepts such as gradients and higher-order derivatives.

Second Order Partial Derivatives

Second order partial derivatives are the derivatives of the first order derivatives. They provide information about the curvature of a function with respect to each variable independently, and when combined, they reveal the local behavior of the function, such as concavity and inflection points.

Mixed Partial Derivatives

Mixed partial derivatives are the second order derivatives taken with respect to different variables. They assess how the rate of change with respect to one variable is affected by changes in another variable. Under appropriate regularity conditions, these mixed derivatives are equal regardless of the order of differentiation.

Hessian Matrix Determinant

The Hessian matrix is a square matrix composed of all second order partial derivatives of a function. Its determinant, often referred to as the Hessian determinant or the discriminant, is used in several applications, such as determining the local curvature of a function and performing second derivative tests to classify critical points in optimization problems.

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