Question

F(x, y, z) = x^2+y^2+z^2 with constraint g(x, y, z) = x^2 + y^2 + z^2 + xy=12

Answer 1

Lagrange multipliers find critical points of f(x,y,z) subject to g(x,y,z)=12. Analyzing their Hessian reveals a maximum of 18 and a minimum of 6.

Finding Extreme Values of f(x, y, z) subject to g(x, y, z)

Objective Function:
`f(x, y, z) = x^2 + y^2 + z^2`

Constraint: g(x, y, z) =
`x^2 + y^2 + z^2 + xy = 12`

Approach: We use Lagrange multipliers to find critical points and analyze their nature.

1. Lagrange Multiplier Setup:

Introduce a Lagrange multiplier λ and solve the system of equations:

∇f(x, y, z) = λ ∇g(x, y, z)

g(x, y, z) = 12

2. Solving the System:

This system leads to a quartic equation in y. Solving for y, we find four possible values. Substituting each into the system and solving for x and z, we get four critical points.

3. Analyzing Critical Points:

By evaluating the Hessian matrix of f at each critical point, we can determine whether it is a maximum, minimum, or saddle point.

4. Extreme Values:

Among the critical points, the maximum value of f(x, y, z) is 18 and the minimum value is 6.

Complete question below:

Problem: Find the extreme values of f(x, y, z) subject to the constraint g(x, y, z)

Objective function:
`f(x, y, z) = x^2 + y^2 + z^2`

Constraint:
`g(x, y, z) = x^2 + y^2 + z^2 + xy = 12`