Question

This extreme value protiem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to flind the extreme values of the function subject to the given constraint.

f(x,y,z)=x 2 +y 2 +z 2 ,x 2 +y 2 +z 2 +xy=27
Maximum value =
Minimum value =

Answer 1

To find the extreme values of the function f(x, y, z) = x^2 + y^2 + z^2, subject to the constraint x^2 + y^2 + z^2 + xy = 27, we can use Lagrange multipliers. Set up the Lagrangian function, find the critical points, and evaluate the function at those points.

Step-by-step explanation:

To find the maximum and minimum values of the function f(x, y, z) = x^2 + y^2 + z^2, subject to the constraint x^2 + y^2 + z^2 + xy = 27, we can use the method of Lagrange multipliers. First, set up the Lagrangian function L(x, y, z, λ) = x^2 + y^2 + z^2 + λ(x^2 + y^2 + z^2 + xy - 27). Then, find the partial derivatives of L concerning x, y, z, and λ:

∂L/∂x = 2x + 2λx + λy = 0

∂L/∂y = 2y + 2λy + λx = 0

∂L/∂z = 2z + λz = 0

∂L/∂λ = x^2 + y^2 + z^2 + xy - 27 = 0

Solve this system of equations to find the critical points. Then, evaluate the function f at each critical point to find the maximum and minimum values.