Question

The function f(x,y,z) = 2x+z² has an absolute maximum value and absolute minimum value subject to the constraint x² +3y² +2z² = 25. Use Lagrange multiples to find these values.

Answer 1

To find the absolute maximum and minimum values of the function with the given constraint, one must use Lagrange multipliers to set up and solve the Lagrange function, subsequently evaluating the critical points in the original function.

Step-by-step explanation:

The problem involves finding the absolute maximum and minimum values of the function f(x, y, z) = 2x + z² subject to the constraint x² + 3y² + 2z² = 25. To solve this, we use Lagrange multipliers.

We start by setting up the Lagrange function L(x, y, z, λ) = 2x + z² - λ(x² + 3y² + 2z² - 25). We then take the partial derivatives of L with respect to x, y, z, and λ, and set them equal to zero. Solving these equations will give us the critical points, which we then evaluate in the original function f to find the maximum and minimum values.