# Find the maximum and minimum values of the function f(x,y,z) = 3x - y - 3z subject to the...

## Question:

Find the maximum and minimum values of the function {eq}f(x,y,z) = 3x - y - 3z {/eq} subject to the constraints {eq}x^2 + 2z^2 = 36 {/eq} and {eq}x + y - z = 4 {/eq}.

## Using Lagrange Multipliers to Find Maximum and Minimum:

In this problem, we combine the objective function, {eq}f(x,y,z) {/eq}, with the constraint functions {eq}g_1(x,y,z) = k {/eq} and {eq}g_2(x,y,z) = j {/eq} into a new function of the form {eq}F(x,y,z,\lambda_1,\lambda_2) = f(x,y,z) - \lambda_1(g_1(x,y,z)-k) - \lambda_2(g_2(x,y,z) - j) {/eq}. Once we have the new function, we take the partial derivatives, set each equal to {eq}0 {/eq} and solve the resulting system of equations. The solutions to the system of equations are the critical points of the function. To determine which is a maximum or minimum, we evaluate each point using the objective function and compare the outputs.

We first combine the objective function and constraint functions into a single function.

{eq}F(x,y,z,\lambda_1,\lambda_2) = f(x,y,z) -...

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