Question

Use Lagrange multipliers to find the maximum and minimum values of f(x,y,z)=x2-8y+10z2 subject to the constraint x2+y2+z2=1

Answer 1

To find the maximum and minimum values of f(x, y, z) = x^2 - 8y + 10z^2 subject to x^2 + y^2 + z^2 = 1, use Lagrange multipliers.

Step-by-step explanation:

To find the maximum and minimum values of the function f(x, y, z) = x^2 - 8y + 10z^2 subject to the constraint x^2 + y^2 + z^2 = 1, we can use Lagrange multipliers.

1. Write down the objective function and the constraint equation:
2. f(x, y, z) = x^2 - 8y + 10z^2
3. g(x, y, z) = x^2 + y^2 + z^2 - 1
4. Compute the gradient of both functions:
5. ∇f(x, y, z) = (2x, -8, 20z)
6. ∇g(x, y, z) = (2x, 2y, 2z)
7. Set up the system of equations:
8. 2x = λ(2x), -8 = λ(2y), 20z = λ(2z), x^2 + y^2 + z^2 = 1
9. Solve the system of equations to find the values of x, y, z, and λ.
10. Substitute the values of x, y, and z into the objective function to find the maximum and minimum values of f(x, y, z).