91Ó°ÊÓ

In calculus, partial derivatives represent the rate of change of a function with respect to one of its variables, holding the other variables constant. For example, if we have a function \( F(x, y, z) = 2x^2 + 3y^2 + z^2 + xy + xz - 2y \), we can find the partial derivatives concerning each variable:
In this function, to find the partial derivative with respect to \( x \), denoted as \( F_x \), we treat \( y \) and \( z \) as constants and differentiate:
\ F_x = \frac{\partial}{\partial x}(2x^2 + 3y^2 + z^2 + xy + xz - 2y) = 4x + y + z. \ Correspondingly for variables \( y \) and \( z \), we get:
\ F_y = \frac{\partial}{\partial y}(2x^2 + 3y^2 + z^2 + xy + xz - 2y) = 6y + x - 2, \
\ F_z = \frac{\partial}{\partial z}(2x^2 + 3y^2 + z^2 + xy + xz - 2y) = 2z + x. \
These partial derivatives help in identifying critical points by being set to zero.

A system of equations is a collection of multiple equations with common variables. To find the minimum point of the function described, we set all partial derivatives to zero and form a system of equations:
1) \( 4x + y + z = 0 \)
2) \( 6y + x - 2 = 0 \)
3) \( 2z + x = 0 \).
We solve these equations simultaneously to find the values of the variables that make all the partial derivatives zero.
Start by solving one equation for one variable, for example, from (3):
\ z = -\frac{x}{2} \
Then substitute \(z\) in equations (1) and (2). This method helps us simplify and systematically find the values for \( x \), \( y \), and \( z \).
Solving it step by step yields:
\ x = -\frac{1}{10}, \
\ y = \frac{7}{20}, \
\ z = \frac{1}{20}. \
The calculated values represent the critical point at \( \left( -\frac{1}{10}, \frac{7}{20}, \frac{1}{20} \right) \).

The Hessian Matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It provides information about the local curvature of the function. For the given function \( F(x, y, z) \), the Hessian Matrix \(H\) is formed as follows:
\ H = \begin{pmatrix} F_{xx} & F_{xy} & F_{xz} \ F_{yx} & F_{yy} & F_{yz} \ F_{zx} & F_{zy} & F_{zz} \ \end{pmatrix} \
To find the second-order partial derivatives, we have:
\ F_{xx} = 4, \
\ F_{yy} = 6, \
\ F_{zz} = 2, \
\ F_{xy} = 1, \
\ F_{xz} = 1, \
\ F_{yz} = 0. \
The Hessian matrix is thus:
\ \begin{pmatrix} 4 & 1 & 1 \ 1 & 6 & 0 \ 1 & 0 & 2 \ \end{pmatrix} \
This matrix is used to analyze the convexity or concavity of the critical points determined earlier.

Critical Points are the values of the variables where all first-order partial derivatives of a function are zero. These points can be minima, maxima, or saddle points.
To confirm if the critical point is a local minimum, we use the Hessian matrix. Compute the determinant of each leading principal minor of the Hessian matrix.
For the Hessian Matrix \(H\):
\ \text{Minor 1: } 4 > 0, \
\ \text{Minor 2 (2x2): } \det(\begin{pmatrix} 4 & 1 \ 1 & 6 \ \end{pmatrix}) = 23 > 0, \
\ \text{Minor 3 (3x3): } \det(H) = 44 > 0. \
Since all principal minors have positive determinants, the critical point \( \left( -\frac{1}{10}, \frac{7}{20}, \frac{1}{20} \right) \) is a local minimum of the function \( F(x, y, z) \).