Problem 32 - Calculus an Applied Approach

Question

Find the critical points of the function and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point. $$ f(x, y, z)=6-[x(y+2)(z-1)]^{2} $$

Step-by-Step Solution

Answer

The critical points of the function \(f(x, y, z)=6-[x(y+2)(z-1)]^{2}\) occur when x=0, y=-2, or z=1 and from the form of the function, we can determine that a relative maximum occurs at each point.

Step 1: Finding the partial derivatives

The partial derivatives of \(f(x, y, z)=6-[x(y+2)(z-1)]^{2}\) are:\[f_x = -2x(y+2)(z-1)^2\]\[f_y = -2x^2(y+2)(z-1)\]\[f_z = -2x^2(y+2)^2(z-1)\]

Step 2: Setting the partial derivatives equal to zero

Setting each of the above equal to zero gives a system of equations to solve:\[-2x(y+2)(z-1)^2 = 0\]\[-2x^2(y+2)(z-1) = 0\]\[-2x^2(y+2)^2(z-1) = 0\]From these equations, we can see that the critical points occur when x=0, y=-2, or z=1.

Step 3: Second Partial Derivative Test

The second partial derivative test is an analytical tool used to determine if a given critical point is a relative maximum, relative minimum, or saddle point. The second partial derivatives are as follows: \[f_{xx} = -2(y+2)^2(z-1)^2 \]\[f_{yy} = -2x^2(z-1)\]\[f_{zz} = -2x^2(y+2)^2\]Performing the second partial derivative test (calculating the determinant of the Hessian matrix), we can determine the nature of each critical point. In this case, as the function \(f(x, y, z)=6-[x(y+2)(z-1)]^{2}\) is a decreasing function in respect to \(x^2\) we can indicate that the function will have a maximum point at each critical point.

Key Concepts

Partial Derivatives

When dealing with functions of multiple variables, it's crucial to understand how changes in one variable can impact the function while the other variables remain constant. This is where partial derivatives come into play. For a function \( f(x, y, z) \), the partial derivative with respect to \( x \), denoted as \( f_x \), investigates how the function changes if only \( x \) is varied. Similarly, \( f_y \) and \( f_z \) determine the changes when \( y \) and \( z \), respectively, are varied. In our original function \( f(x, y, z)=6-[x(y+2)(z-1)]^2 \), we compute these partial derivatives as:

• \( f_x = -2x(y+2)(z-1)^2 \)
• \( f_y = -2x^2(y+2)(z-1) \)
• \( f_z = -2x^2(y+2)^2(z-1) \)

These expressions show the sensitivity of the function to changes in \( x \), \( y \), and \( z \). Setting these partial derivatives to zero helps us find critical points, which are potential candidates for local maxima, minima, or saddle points.

Second Partial Derivative Test

Once we have identified critical points using partial derivatives, the next step is to determine their nature. That's where the second partial derivative test comes into play. This test involves using the second derivatives of the function to analyze the concavity around a critical point. For a function \( f(x, y, z) \), we calculate second derivatives such as \( f_{xx} \), \( f_{yy} \), and \( f_{zz} \). In our specific case, these were:

• \( f_{xx} = -2(y+2)^2(z-1)^2 \)
• \( f_{yy} = -2x^2(z-1) \)
• \( f_{zz} = -2x^2(y+2)^2 \)

The second partial derivative test checks the sign of the second derivatives and involves computing determinants to conclude if a critical point is a local maximum, minimum, or a saddle point. However, caution is required as the test does not apply all the time, particularly if determinants are zero, indicating an inconclusive result.

Hessian Matrix

The Hessian matrix is a powerful tool for the second partial derivative test. It is a square matrix of second-order partial derivatives of a scalar-valued function. For our function \( f(x, y, z) \), the Hessian matrix \( H \) is structured as follows:\[ H = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \ f_{yx} & f_{yy} & f_{yz} \ f_{zx} & f_{zy} & f_{zz} \end{bmatrix} \]For our exercise, we have calculated some of these second derivatives, like \( f_{xx} \), \( f_{yy} \), and \( f_{zz} \). In practice, one would also need to calculate mixed derivatives like \( f_{xy}, f_{xz}, \) and so forth, to complete the matrix.The determinant of this Hessian matrix at a critical point helps us decide on the concavity or the type of critical point for the function. A positive determinant at a point suggests the point is a local extremum, whereas a negative determinant indicates a saddle point. The Hessian matrix serves as a comprehensive tool combining partial derivatives to provide clear insights into the behavior of the function at critical points.