Question

Apply the method of Lagrange multipliers to the function f(x,y)=(x2+1)y subject to the constraint x2+y2=29. Hint: First, show that y=0. Then treat the cases x=0 and x=0 separately. (Use decimal notation. Give your answers to two decimal places.) maximum: Incorrect minimum

Answer 1

To maximize or minimize a function using Lagrange multipliers, form the Lagrange function including the constraint, take partial derivatives, set them to zero, and solve the resulting system of equations.

Step-by-step explanation:

To apply the method of Lagrange multipliers to the function f(x,y)=(x2+1)y, with the constraint x2+y2=29, we first state the Lagrange function L(x, y, λ) = f(x, y) - λ(g(x, y) - c), where g(x, y) is the constraint function and c is the constraint target value.

In this case, L(x, y, λ) = (x2+1)y - λ(x2+y2-29).

We need to take partial derivatives of L with respect to x, y, and λ and set them equal to zero:

• ∂L/∂x = 2xy - 2λx = 0
• ∂L/∂y = x2 + 1 - 2λy = 0
• ∂L/∂λ = 29 - (x2 + y2) = 0

We then solve the system of equations to find critical points, taking into account the hint that y ≠ 0, and considering the cases where x = 0 and x ≠ 0 separately. By solving these equations, we can find the values that maximize or minimize the function f(x, y) under the given constraint.