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The Hessian determinant is a crucial concept in multivariable calculus, particularly when analyzing critical points of functions with more than one variable. It is a determinant of a matrix called the Hessian matrix, which consists of the second partial derivatives of a function. This matrix gives insight into the curvature of the function at a given point.
More specifically:

• For a function of two variables, such as \( f(x, y) \), the Hessian matrix is a 2x2 matrix, given by:

\[H = \begin{bmatrix}\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2}\end{bmatrix}\]
The Hessian determinant at a point \((x_0, y_0)\) is then:\[\Delta(x_0, y_0) = \frac{\partial^2 f}{\partial x^2} \frac{\partial^2 f}{\partial y^2} - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2\]
The determinant helps determine the nature of a critical point:

• If \( \Delta > 0 \), and \( \frac{\partial^2 f}{\partial x^2} > 0 \), the point is a local minimum.
• If \( \Delta > 0 \), and \( \frac{\partial^2 f}{\partial x^2} < 0 \), the point is a local maximum.
• If \( \Delta < 0 \), the point is a saddle point, indicating a change in concavity.
• If \( \Delta = 0 \), the test is inconclusive, and further analysis is required.

In the given function, the Hessian determinant \( \Delta(0,0) = 0 \), suggesting an inconclusive result, leading us to verify other behavior trends of the function to categorize the critical point.

Partial derivatives are foundational in studying functions of multiple variables. They represent the rate of change of a function in relation to one of the variables, while keeping others constant.
For a function \( f(x, y) \):

• The partial derivative with respect to \( x \), \( \frac{\partial f}{\partial x} \), measures how the function changes as \( x \) changes, keeping \( y \) constant.
• The partial derivative with respect to \( y \), \( \frac{\partial f}{\partial y} \), measures the change of the function as \( y \) varies, with \( x \) held fixed.

A higher-order partial derivative is simply a derivative of a partial derivative. In the context of analyzing critical points:

• The first-order partial derivatives are set to zero to find critical points.
• The second-order partial derivatives are used to decipher the nature of these critical points using Hessian Determinant analysis.

In our example, these derivatives were used to calculate critical points and to evaluate the behavior of the function at these points.

Critical points are specific locations in the domain of a function where the first partial derivatives equal zero. These points are essential as they potentially indicate local maxima, minima, or saddle points.
To find critical points for \( f(x, y) = x^3 y^3 \):

• Calculate the first-order partial derivatives \( \frac{\partial f}{\partial x} = 3x^2 y^3 \) and \( \frac{\partial f}{\partial y} = 3x^3 y^2 \).
• Set each derivative to zero and solve for \( x \) and \( y \).
• In this function, \( x = 0 \) and \( y = 0 \) satisfy both conditions. Thus, a critical point is found at \((0,0)\).

Once identified, the nature of the critical point must be determined. This is often done using the second derivative test, which involves:

• Calculating the Hessian determinant to infer local behavior around the point.
• Analyzing the results in context—in this case, \( \Delta(0,0) = 0 \), indicating a saddle point by further functional investigation.

Understanding critical points is vital for exploring the topography of a multivariable function and predicting its behavior.