Question

If \(f_{x}(a, b)=f_{y}(a, b)=0,\) does it follow that \(f\) has a local maximum or local minimum at \((a, b) ?\) Explain.

Short answer

Short Answer: Based on the given conditions \(f_x(a, b) = f_y(a, b) = 0\), we cannot explicitly determine whether \((a, b)\) is a local maximum or minimum for function \(f\). We need to further analyze the determinant of the Hessian matrix and the second partial derivatives to determine the nature of the critical point.

Step-by-step solution

Identify the critical point

First, we are given that the partial derivatives of the function \(f\) are zero at \((a, b)\), i.e., \(f_x(a, b) = 0\) and \(f_y(a, b) = 0\). This means that \((a, b)\) is a critical point of the function \(f\). Critical points are the potential candidates for local minima, maxima, or saddle points.

Determine the conditions for local maxima and minima

A point \((a, b)\) is a local maximum or minimum if the second partial derivatives of \(f\) satisfy certain conditions. To determine these conditions, we compute the second partial derivatives \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\), and analyze the determinant of the Hessian matrix.

Hessian Matrix

Create a Hessian matrix \(H\) for the function \(f\): $$ H = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} $$

Find the determinant of Hessian Matrix

The determinant of a 2x2 matrix is given by: $$ \text{det}(H) = f_{xx}(a, b)\cdot f_{yy}(a, b) - f_{xy}(a, b)\cdot f_{yx}(a, b) $$

Analyze det(H) for local maxima and minima

Depending on the determinant of the Hessian matrix, we can make the following conclusions: 1. If \(\text{det}(H) > 0\) and \(f_{xx}(a, b) > 0\), \((a, b)\) is a local minimum. 2. If \(\text{det}(H) > 0\) and \(f_{xx}(a, b) < 0\), \((a, b)\) is a local maximum. 3. If \(\text{det}(H) < 0\), \((a, b)\) is a saddle point. 4. If \(\text{det}(H) = 0\), the test is inconclusive, and further analysis is required. From the given conditions \(f_x(a, b) = f_y(a, b) = 0\), we cannot explicitly determine whether \((a, b)\) is a local maximum or minimum, as the result depends on the Hessian determinant and the sign of the second partial derivative \(f_{xx}(a, b)\).

Key concepts

Critical Points

A critical point is a set of values where the first partial derivatives of a function are equal to zero. For a multivariable function, such as one with inputs \((x,y)\), the critical point \((a, b)\) is determined when both partial derivatives equal zero: \(f_x(a, b) = 0\) and \(f_y(a, b) = 0\). This means that the slope of the function is flat at that point in all directions, making it a candidate for local extrema or a saddle point.
A critical point doesn't automatically mean a function has a local maximum or minimum there. It's just an indicator that the function's behavior could change at that point. In practice, critical points help us focus on smaller areas to determine if a significant property like a maximum, minimum, or saddle occurs.

Second Partial Derivatives

To better understand the character of the critical point, the second partial derivatives are very useful. These derivatives provide information on how the slope changes, offering clues about the function's concavity at the critical point.
We compute these second partial derivatives: \(f_{xx}(a, b)\), \(f_{yy}(a, b)\), and \(f_{xy}(a, b) = f_{yx}(a, b)\). These values build up the framework for analyzing the function's behavior around the point \((a, b)\).
Second partial derivatives help form the Hessian matrix, which is a tool for deeper analysis. They help us understand whether the region around the critical point is curving upwards or downwards, aiding in finding whether a point is a maximum, minimum, or saddle.

Local Maxima and Minima

Local maxima and minima refer to points where a function reaches its highest or lowest value relative to the surrounding points. Determining these requires carefully examining the critical points and their surrounding behavior.
At a local maximum, the function value at the critical point is higher than its immediate neighborhood. Conversely, a local minimum is where the function is lower than its surroundings. For multivariable functions, identifying these moments involve assessing how the surface of the function behaves around \((a, b)\).
One pivotal method is using the conditions derived from second order derivatives and determinant analysis to conclude about whether \((a, b)\) is a peak, a trough, or neither.

Determinant Analysis

Determinant analysis involves evaluating the determinant of the Hessian matrix to understand the nature of the critical point. The Hessian matrix \(H\) is constructed using the second partial derivatives and is given by: \[ H = \begin{bmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy} \end{bmatrix} \] The determinant of this two-by-two matrix is calculated by \(\text{det}(H) = f_{xx}(a, b) \cdot f_{yy}(a, b) - f_{xy}(a, b) \cdot f_{yx}(a, b)\).
Depending on the value of this determinant, conclusions can be drawn:

• If \( \text{det}(H) > 0 \) and \( f_{xx} > 0 \), the point is a local minimum.
• If \( \text{det}(H) > 0 \) and \( f_{xx} < 0 \), the point is a local maximum.
• If \( \text{det}(H) < 0 \), the point is a saddle point.
• If \( \text{det}(H) = 0 \), the test isn't definitive.

Thus, determinant analysis of the Hessian helps pinpoint the exact behavior of critical points.