91Ó°ÊÓ

Partial derivatives are a fundamental tool in multivariable calculus that help us understand how a function changes with respect to one of its variables, while keeping the other variables constant. For a function of three variables, such as \(f(x_1, x_2, x_3)\), we compute the partial derivatives with respect to \(x_1, x_2,\) and \(x_3\) to explore the function's behavior. Specifically, we find the partial derivatives:

• \(f_{x_1} = \frac{\partial f}{\partial x_1}\)
• \(f_{x_2} = \frac{\partial f}{\partial x_2}\)
• \(f_{x_3} = \frac{\partial f}{\partial x_3}\)

These derivatives are set to zero to locate the stationary points, where the function's slope in each direction is zero. For the given function \(f(x_1, x_2, x_3) = x_1^2 + x_2^2 + 3x_3^2 - x_1x_2 + 2x_1x_3 + x_2x_3\), the partial derivatives were computed as:

• \(f_{x_1} = 2x_1 - x_2 + 2x_3\)
• \(f_{x_2} = 2x_2 - x_1 + x_3\)
• \(f_{x_3} = 6x_3 + 2x_1 + x_2\)

This is the first step in finding the stationary points.

A stationary point is where the partial derivatives of a function are zero. This means the function does not change in any direction from this point. To find a stationary point, we solve the system of equations obtained by setting the partial derivatives to zero:

• \(2x_1 - x_2 + 2x_3 = 0\)
• \(2x_2 - x_1 + x_3 = 0\)
• \(6x_3 + 2x_1 + x_2 = 0\)

Solving this system provides the stationary point for our function, which is \((0, 0, 0)\). The next step is to determine the nature of this point by examining the second derivatives.

The Hessian matrix is a square matrix of second-order partial derivatives of a function. It provides essential information about the curvature of the function at a stationary point. For a function of three variables, the Hessian matrix \(H\) is:
\[ H = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \frac{\partial^2 f}{\partial x_1 \partial x_3} \ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \frac{\partial^2 f}{\partial x_2 \partial x_3} \ \frac{\partial^2 f}{\partial x_3 \partial x_1} & \frac{\partial^2 f}{\partial x_3 \partial x_2} & \frac{\partial^2 f}{\partial x_3^2} \end{bmatrix} \]
For our function, the Hessian matrix is:
\ H = \begin{bmatrix} 2 & -1 & 2 \ -1 & 2 & 1 \ 2 & 1 & 6 \end{bmatrix} \
We use the Hessian matrix to determine if a stationary point is a local minimum, maximum, or saddle point.

A matrix is positive definite if all its leading principal minors are positive. This property is crucial in establishing whether a stationary point is a local minimum. For our Hessian matrix, we calculate the leading principal minors:

• First minor: \(D_1 = 2\)
• Second minor: \(D_2 = \begin{vmatrix} 2 & -1 \ -1 & 2 \end{vmatrix} = 3\)
• Third minor: \(D_3 = \begin{vmatrix} 2 & -1 & 2 \ -1 & 2 & 1 \ 2 & 1 & 6 \end{vmatrix} = 35\)

All these determinants are positive, confirming that the Hessian matrix is positive definite. Therefore, the stationary point \((0, 0, 0)\) is a local minimum for our function. Understanding positive definiteness helps in identifying the nature of stationary points accurately.