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. $$ f(x, y, z)=6-[x(y+2)(z-1)]^{2} $$

Short answer

The critical points of the function are obtained by solving the system of equations resulting in points (0,-2,1). Examining the determinant of the Hessian matrix at each point will reveal whether these points are local minima or maxima.

Step-by-step solution

Find the Partial Derivatives

The first step involves determining the partial derivatives for the given function with respect to the variables x, y and z. Applying the chain rule, we get: \[ \frac{\partial f}{\partial x} = 2x(y+2)(z-1)\] \[ \frac{\partial f}{\partial y} = 2x^{2}(z-1)\] \[ \frac{\partial f}{\partial z} = -2x^{2}(y+2)\]

Solve equations for critical points

The next step is to set each of these equations equal to zero and solve the system of equations to find critical points. This result in \(x=0\) or \(y=-2\) or \(z=1\).

Check for relative maximum or minimum

To check whether each point is a relative maximum or minimum, we compute the second order partial derivatives and form the Hessian matrix. We then examine the determinant of this matrix at each critical point to determine the nature of the critical point. If it is positive, we have a local minimum, if it is negative, we have a local maximum.