91Ó°ÊÓ

Understanding partial derivatives is essential when dealing with functions of multiple variables. They help us determine how a function changes as we vary one of its variables while keeping the other variables constant. For a function of two variables, say \( f(a, b) \), the partial derivative with respect to \( a \), denoted as \( f_a \), tells us the rate of change of \( f \) as only \( a \) changes. Similarly, the partial derivative with respect to \( b \), denoted as \( f_b \), informs us of the rate of change in \( f \) as \( b \) changes.

In our example, the function is \( f(a, b) = a^2 - 4a + b^2 - 2b - 12 \). Calculating partial derivatives for \( a \) and \( b \):

• \( f_a(a, b) = 2a - 4 \)
• \( f_b(a, b) = 2b - 2 \)

These partial derivatives are crucial because they help us find critical points where the function might have a local minimum, maximum, or a saddle point, by setting them to zero and solving for \( a \) and \( b \).

Once we find critical points using partial derivatives, it's crucial to determine their nature. The second derivative test is a handy tool for this. It tells us whether each critical point is a local minimum, maximum, or a saddle point.

The second derivative test involves computing the second partial derivatives:

• \( f_{aa} \), the second derivative with respect to \( a \)
• \( f_{bb} \), the second derivative with respect to \( b \)
• \( f_{ab} \), the mixed partial derivative

For the function \( f(a, b) = a^2 - 4a + b^2 - 2b - 12 \), these are:

• \( f_{aa} = 2 \)
• \( f_{bb} = 2 \)
• \( f_{ab} = 0 \)

The critical point \((2, 1)\) is evaluated by calculating the discriminant \( D = f_{aa}f_{bb} - (f_{ab})^2 \). If \( D > 0 \) and \( f_{aa} > 0 \), the point is a local minimum.

The Hessian matrix plays a pivotal role when utilizing the second derivative test. It is a square matrix composed of second-order partial derivatives of a function. For a function of two variables, the Hessian matrix typically appears as:\[H = \begin{bmatrix}f_{aa} & f_{ab} \f_{ab} & f_{bb}\end{bmatrix}\]

In our exercise, the function's Hessian matrix is:\[H = \begin{bmatrix}2 & 0 \0 & 2\end{bmatrix}\]

The determinant of this matrix, \( D = f_{aa} f_{bb} - (f_{ab})^2 \), helps classify the critical point. It evaluates how the curvature of the function behaves at that point. A positive determinant with \( f_{aa} > 0 \) at the critical point signifies a local minimum. Thus, in our example, since \( D = 4 > 0 \) and \( f_{aa} = 2 > 0 \), the point \((2, 1)\) is confirmed to be a local minimum.