Question

Find the maximum and minimum values of\(f\)subject the given constraints. Use a computer algebra system to solve system of equations that arises in using Lagrange multiple. (If your CAS finds only one solution, you may need to additional commands.)\(f(x,y,z) = x + y + z;\;\;\;{x^2} - {y^2} = z,{x^2} + {z^2} = 4\)

Short answer

Minimum value is \( - 4.74\) & maximum value is\(3.99\).

Step-by-step solution

Method of Lagrange multipliers

To find the maximum and minimum values of\(f(x,y,z)\)subject to the constraint\(g(x,y,z) = k\)(assuming that these extreme values exist and\(\nabla g \ne {\bf{0}}\)on the surface\(g(x,y,z) = k)\):

(a) Find all values of\(x,y,z\), and\(\lambda \)such that\(\nabla f(x,y,z) = \lambda \nabla g(x,y,z)g(x,y,z) = k\)

And

(b) Evaluate\(f\)at all the points\((x,y,z)\)that result from step (a). The largest of these values is the maximum value of\(f\); the smallest is the minimum value of\(f\).

Given data

Consider\(f(x,y,z) = x + y + z;\;\;\;{x^2} - {y^2} = z,\;\;\;{x^2} + {z^2} = 4\).

Lagrange multipliers are used to optimize functions subject to two constraints. Using \(g(x,y,z) = {x^2} - {y^2} - z\)and \(h(x,y,z) = {x^2} + {z^2}\)the constraints can be expressed in the form \(g = 0\)and\(h = 4\).

The system of equations to solve has the form\(\nabla f = \lambda \nabla g + \mu \nabla h\)

Form equation

Use these partials to set up the Lagrange multiplier system of equations.

\(\begin{aligned}{c}1 &= 2\lambda x + 2\mu x\\1 &= - 2\lambda y\\1 &= - \lambda + 2\mu z\\{x^2} - {y^2} - z &= 0\\{x^2} + {z^2} &= 4\end{aligned}\)

Using calculator find maximum & minimum

Evaluate \(f\)at each solution point to determine the maximum and minimum values. The table below gives the numerical approximations to the solutions and the function value.