Question

Find the minimum and maximum values of the function ( f(x, y, z)=9 x+2 y+2 z subject to the constraint x²+2 y²+3 z²=1

Answer 1

To find the minimum and maximum of f(x, y, z) = 9x + 2y + 2z given the constraint x² + 2y² + 3z² = 1, use the method of Lagrange multipliers. Solve the system of equations that comes from the gradients of the function and the constraint for critical points, then evaluate f at those points.

Step-by-step explanation:

To find the minimum and maximum values of the function f(x, y, z) = 9x + 2y + 2z subject to the constraint x² + 2y² + 3z² = 1, we can use the method of Lagrange multipliers. This involves setting up the system of equations derived from the gradient of the function and the gradient of the constraint multiplied by a Lagrange multiplier λ:

• ∇f(x, y, z) = λ∇g(x, y, z), where g(x, y, z) = x² + 2y² + 3z² - 1 = 0.

We solve these equations to find the critical points which may include the minimum and maximum values:

1. 9 = 2λx
2. 2 = 4λy
3. 2 = 6λz

Using these equations and the constraint, we can solve for x, y, z, and λ. Finally, we evaluate f(x, y, z) at the critical points to determine the minimum and maximum values