Question

Solve the equation 2x + 3y - z - 11 = 0 for which 4x² + y² + z² is a minimum.

a) x = 2, y = -3, z = 1
b) x = -2, y = 3, z = -1
c) x = 1, y = 2, z = -3
d) x = -1, y = -2, z = 3

Answer 1

To solve the equation 2x + 3y - z - 11 = 0 for which 4x² + y² + z² is a minimum, we can use the method of Lagrange multipliers. The values of x = 2, y = -3, and z = 1 satisfy both equations.

Step-by-step explanation:

To solve the equation 2x + 3y - z - 11 = 0 for which 4x² + y² + z² is a minimum, we need to find the values of x, y, and z that satisfy both equations. To do this, we can use the method of Lagrange multipliers.

First, we set up the Lagrange function L = 4x² + y² + z² + λ(2x + 3y - z - 11). Taking the partial derivatives with respect to x, y, z, and λ, we get:

8x + 2λ = 0

2y + 3λ = 0

2z - λ = 0

2x + 3y - z = 11

Solving this system of equations, we find x = 2, y = -3, and z = 1. Therefore, the correct answer is a) x = 2, y = -3, z = 1.