Question

Jse Lagrange multiplier techniques to find the local extreme values of the given function subject to the stated constraint. f(x,y)=e²ˣʸ with constraint g(x,y)=x³ +y³ =16

Answer 1

To find the local extreme values of the function f(x,y)=e²ˣʸ subject to the constraint g(x,y)=x³ +y³ =16 using Lagrange multiplier techniques, set up the Lagrangian function by finding the partial derivatives, solve the resulting system of equations, and substitute the values back into the original function.

Step-by-step explanation:

To find the local extreme values of the function f(x,y)=e²ˣʸ subject to the constraint g(x,y)=x³ +y³ =16 using Lagrange multiplier techniques, follow these steps:

1. Set up the Lagrangian function:
2. L(x,y,λ) = f(x,y) - λ(g(x,y) - 16)
3. Find the partial derivatives of the Lagrangian function:
4. ∂L/∂x = 2e²ˣʸ - 3λx²
5. ∂L/∂y = 2e²ˣʸ - 3λy²
6. ∂L/∂λ = x³ + y³ - 16
7. Set the partial derivatives equal to zero and solve the resulting system of equations:
8. Solving the system will give you values for x, y, and λ.
9. Substitute the values of x, y, and λ back into the original function to find the local extreme values.

For example, if you find that x = 2, y = 2, and λ = 2, substitute these values into the original function to find the local extreme value.