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. $$ f(x, y, z)=4-[x(y-1)(z+2)]^{2} $$

Short answer

The critical points of the function \( f(x, y, z)=4-[x(y-1)(z+2)]^{2} \) are (0,1,-2) and (0, any number, any number). The point (0,1,-2) is a relative maximum and the nature of the point (0, any number, any number) cannot be definitively identified without a choice of specific values for 'any number'.

Step-by-step solution

Calculating Partial Derivatives

Given a multi-variable function, a critical point occurs whenever the partial derivatives of that function with respect to each variable equals to zero.\n Let's calculate the partial derivatives of the given function with respect to each variable (x, y, and z).\nThe partial derivative of \( f \) with respect to \( x \) is: \( f_x = -2x(y-1)(z+2) \).\nThe partial derivative of \( f \) with respect to \( y \) is: \( f_y = -2x^{2}(z+2) \).\nThe partial derivative of \( f \) with respect to \( z \) is: \( f_z = -2x^{2}(y-1) \).

Finding Critical Points

Now we need to solve the following system of equations to find the critical points:\( f_x = 0 \)\( f_y = 0 \)\( f_z = 0 \)\nSolving these equations gives us two critical points (0, 1, -2) and (0, any number, any number) where 'any number' can be any real number.

Identifying Relative Maxima or Minima

To check whether these critical points are maxima, minima, or saddle points, we need to analyze the function’s value in a neighborhood around the critical point. For instance, taking the point (0,1,-2), we can inspect the function's value at a point slightly displaced from this -- say (0+δ1, 1+δ2, -2+δ3) where δ1, δ2, and δ3 are small displacements. This gives \( f(0+δ1, 1+δ2, -2+δ3) = 4 - δ1^2(δ2)(2+δ3)^2 \). As δ1, δ2 and δ3 approaches zero this tends to 4 therefore (0,1,-2) is relative maximum. The point (0, any number, any number) could also be any value depending on the choice of 'any number' values. Thus, we cannot definitively say whether these points are relative maxima or minima.