Question

What is the absolute maximum value and the absolute minimum value? The function f(x,y,z)=(xyz)¹/² has an absolute maximum value and absolute minimum value subject to the constraint x+y+z=3 with x≥​0, y≥​0, and z≥0. Use Lagrange multipliers to find these values. The absolute maximum value is __. The absolute minimum value is ___.

Answer 1

To find the absolute maximum and minimum values of the function f(x,y,z)=(xyz)¹/² subject to the constraint x+y+z=3 with x≥0, y≥0, and z≥0, we can use Lagrange multipliers.

Step-by-step explanation:

The function f(x,y,z)=(xyz)¹⁄₂ has an absolute maximum value and absolute minimum value subject to the constraint x+y+z=3 with x≥​0, y≥​0, and z≥0. To find these values, we can use Lagrange multipliers.

Let's define the objective function g(x,y,z) = (xyz)¹⁄₂ and the constraint function h(x,y,z) = x + y + z - 3. We will solve the system of equations ∂g/∂x = λ∂h/∂x, ∂g/∂y = λ∂h/∂y, and ∂g/∂z = λ∂h/∂z, where λ is the Lagrange multiplier.

Solving these equations will give us the values of x, y, and z that correspond to the absolute maximum and absolute minimum values of f(x,y,z).