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The Hessian matrix is a second-order square matrix derived from the second partial derivatives of a scalar-valued function. It is an essential tool in multivariable calculus used to determine the local curvature of functions, especially in optimization problems.

For a function like \( f(x, y, z) \), the Hessian matrix \( H \) is composed of elements \( f_{xx}, f_{yy}, f_{zz}, \) along with the mixed partial derivatives \( f_{xy}, f_{xz}, \text{and} \; f_{yz} \).
This matrix is pivotal when assessing the nature of critical points, which could be potential local minima, maxima, or saddle points.

The Hessian matrix is a compact way to represent how the function behaves around a specific point, often a critical point, through its second derivative test.

• The diagonal elements represent the second partial derivatives of each variable with respect to itself.
• Off-diagonal elements show the change between different variables, representing cross-partial derivatives.

In our problem, we created the Hessian matrix for \( f(x, y, z) = x^2 + y^2 + z^2 + xy \) which takes the form:\[ H = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \ f_{yx} & f_{yy} & f_{yz} \ f_{zx} & f_{zy} & f_{zz} \end{bmatrix} = \begin{bmatrix} 2 & 1 & 0 \ 1 & 2 & 0 \ 0 & 0 & 2 \end{bmatrix} \]This indicates the interaction between the variables in terms of their second derivatives.

A local minimum of a function is a point where the function value is lower than at any nearby points. This doesn't mean it's the lowest value of the function possible, only that it's lesser in its immediate vicinity.

Mathematically, for a critical point \( \textbf{c} = (x_0, y_0, z_0) \), if the Hessian matrix at this point is positive definite, then the function has a local minimum at \( \textbf{c} \). This is because positive definiteness implies the surface is curving upward in every direction, creating a "bowl" shape.

• For a matrix to be positive definite, all its leading principal minors must be positive.
• In our example, we found that the critical point \((0, 0, 0)\) of the function \(f(x, y, z) = x^2 + y^2 + z^2 + xy\) satisfies these conditions with the Hessian matrix at this point.

We calculated different determinants of submatrices and confirmed that all were positive, indicating a local minimum. This conclusion mirrors how the physical slope behaves around that point, meaning that every direction you move from \((0,0,0)\), the function value increases slightly.

A partial derivative measures how a multivariable function changes as one variable changes, while all other variables are held constant. It's a crucial concept when analyzing multivariable functions, as it helps in understanding how functions behave and interact in different dimensions.

To find critical points, we need to set the first partial derivatives to zero. These are places where the rate of change is zero in each direction, which might suggest a maximum, minimum, or saddle point.

• The function \( f(x, y, z) = x^2 + y^2 + z^2 + xy \) required us to find \( f_x, f_y, \text{and} \; f_z \), the partial derivatives with respect to \(x, y,\) and \(z\).
• Expressions: \( f_x = 2x + y \), \( f_y = 2y + x \), and \( f_z = 2z \).

By equating these partial derivatives to zero, we found the critical points.

Once critical points are identified using partial derivatives, further analysis using higher-order derivatives like in the Hessian matrix provides insight into the nature of these points, guiding us toward understanding whether they're points of minima, maxima, or saddle points.
This foundational step is how we commence investigations into the function's behavior near potential extremal points.