91Ó°ÊÓ

When dealing with multivariable functions, understanding how variables change independently is crucial. Partial derivatives allow us to explore this by examining the rate of change of a function concerning one variable while keeping other variables constant.
For example, given a function \( h(w, z) = 0.6w^2 + 1.3z^3 - 4.7wz \), you can find the partial derivative with respect to \( w \), noted as \( h_w \), by treating \( z \) as a constant. The same principle applies when finding the partial derivative with respect to \( z \), known as \( h_z \).
Calculating partial derivatives helps identify critical points, where the function's slope is temporarily zero. These points occur where \( h_w = 0 \) and \( h_z = 0 \), offering vital information about the function's behavior, like whether it reaches a peak, valley, or saddle point.

The Hessian matrix plays a fundamental role in determining the nature of critical points. It is a square matrix of second-order partial derivatives, structured to provide deeper insight into the behavior around critical points.
For our function, the Hessian matrix \( H \) is constructed from the second-order partial derivatives: \( h_{ww} \), \( h_{zz} \), and \( h_{wz} \). Specifically:

• \( h_{ww} = \frac{\partial^2 h}{\partial w^2} = 1.2 \)
• \( h_{zz} = \frac{\partial^2 h}{\partial z^2} = 7.8z \)
• \( h_{wz} = \frac{\partial^2 h}{\partial w \partial z} = -4.7 \)

The Hessian determinant \( D \) is calculated as \( h_{ww}h_{zz} - (h_{wz})^2 \). This determinant helps us classify the nature of critical points:

• If \( D > 0 \) and all second partial derivatives are positive, the point is a local minimum.
• If \( D > 0 \) and at least one second partial derivative is negative, the point is a local maximum.
• If \( D < 0 \), the point is a saddle point, indicating neither a max nor a min.

To classify the critical points identified using partial derivatives, we employ the Second Derivative Test. This test uses the Hessian matrix to evaluate the nature of these points by examining the concavity of the function in multiple dimensions.
The outcome of this test is reliant on the Hessian determinant \( D \):

• If \( D > 0 \), the point may be an extremum (either maximum or minimum depending on whether \( h_{ww} \) is positive or negative).
• If \( D < 0 \), it indicates a saddle point, a point where the function has traits of both increasing and decreasing tendencies.
• If \( D = 0 \), the test is inconclusive, and further analysis is needed to determine the point's nature.

In our exercise, both critical points \((0,0)\) and \((18.503, 4.7226)\) resulted in a negative Hessian determinant, classifying them as saddle points. This suggests these points possess characteristics of neither absolute highs nor lows but rather mixed behaviors in different directions.