91Ó°ÊÓ

The second derivative test is a crucial tool for analyzing critical points of multivariable functions. It helps determine whether a critical point is a local minimum, local maximum, or a saddle point. This involves examining the second partial derivatives at the critical point. For any function like \[ f(x, y) = 3x^2 + ky^2 + 9xy \], we first find the second derivatives \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \).
The next step involves calculating the Hessian matrix determinant, \( D \), which is given by: \[ D = f_{xx} f_{yy} - (f_{xy})^2 \].
The sign of \( D \) will tell us a lot:

• If \( D > 0 \) and \( f_{xx} > 0 \), the point is a local minimum.
• If \( D > 0 \) and \( f_{xx} < 0 \), the point is a local maximum.
• If \( D < 0 \), the point is a saddle point.

A critical point where \( D = 0 \) is inconclusive, and further analysis is needed to classify the point.

The Hessian matrix is central to understanding the behavior around critical points of a multivariable function. It is a square matrix of second-order partial derivatives. For the function \( f(x, y) = 3x^2 + ky^2 + 9xy \), its Hessian matrix \( H \) is:\[ H = \begin{bmatrix} f_{xx} & f_{xy} \ f_{xy} & f_{yy} \end{bmatrix} = \begin{bmatrix} 6 & 9 \ 9 & 2k \end{bmatrix} \]
The determinant of the Hessian matrix at any point, denoted \( D \), is crucial for the second derivative test. This determinant, \( D = f_{xx} f_{yy} - (f_{xy})^2 \), assesses whether the Hessian matrix is positive definite, negative definite, or indefinite:

• A positive definite Hessian (\( D > 0 \) and \( f_{xx} > 0 \)) suggests a local minimum.
• A negative definite Hessian (\( D > 0 \) and \( f_{xx} < 0 \)) indicates a local maximum.
• An indefinite Hessian (\( D < 0 \)) reveals a saddle point.

Understanding how to use the Hessian effectively allows you to easily classify critical points.

Partial derivatives measure how a function changes as one of the variables changes while all other variables are held constant. For a function \( f(x, y) \), the partial derivative with respect to \( x \) is noted as \( f_x \), and with respect to \( y \), it’s \( f_y \). For the function \( f(x, y) = 3x^2 + ky^2 + 9xy \), the partial derivatives are:

• \( f_x = 6x + 9y \)
• \( f_y = 2ky + 9x \)

These derivatives form the gradient vector of the function. At critical points, both partial derivatives equal zero, indicating points where the function may have a local extremum or saddle point. Solving:

• \( 6x + 9y = 0 \)
• \( 9x + 2ky = 0 \)

provides the critical points. This analysis is foundational to applying the second derivative test.

Local extremum classification helps us identify whether a critical point of a function is a local maximum, minimum, or a saddle point. Using the second derivative test and the Hessian matrix, we can carefully classify the nature of these points.
For instance, in the function \( f(x, y) = 3x^2 + ky^2 + 9xy \), once the critical point \( (0,0) \) is found, examining the Hessian determinant \( D = 12k - 81 \) provides clues for classification:

• A local minimum exists when \( k > \frac{27}{4} \) when \( D > 0 \) and \( f_{xx} > 0 \).
• A local maximum is not possible due to \( f_{xx} > 0 \), no k satisfies \( f_{xx} < 0 \).
• A saddle point is present when \( k < \frac{27}{4} \) where \( D < 0 \).

Such classification allows you to understand the behavior of the function near the critical points, demonstrating how manipulating parameters like \( k \) can shift the nature of these points.