Question

Find the maximum and minimum values of f(x, y, z) = x - 2y + 5z on the sphere x² + y² + z² = 30.

Answer 1

To find the maximum and minimum values of the function f(x, y, z) = x - 2y + 5z on the sphere x² + y² + z² = 30, we can use the method of Lagrange multipliers.

Step-by-step explanation:

To find the maximum and minimum values of the function f(x, y, z) = x - 2y + 5z on the sphere x² + y² + z² = 30, we can use the method of Lagrange multipliers. Here are the steps:

1. Set up the Lagrange equation: L(x, y, z, λ) = x - 2y + 5z + λ(x² + y² + z² - 30)
2. Take the partial derivatives of L with respect to x, y, z, and λ.
3. Set the partial derivatives equal to zero and solve for x, y, z, and λ.
4. Substitute the values of x, y, and z back into the original function f(x, y, z) to find the maximum and minimum values.

By following these steps, you will be able to determine the maximum and minimum values of the function f(x, y, z) on the given sphere.

We'll set up the function L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - 30) where g(x, y, z) is the constraint x² + y² + z². We find the partial derivatives of L with respect to x, y, z, and λ and set them equal to zero to solve for x, y, z, and λ which provide the critical points. Then, we evaluate f at these critical points to determine the maximum and minimum values on the sphere.