When maximizing a function f(x,y) subject to a constraint g(x,y)=a, which of the

Question

When maximizing a function f(x,y) subject to a constraint g(x,y)=a, which of the following are true?

Question 1 options:

The max and min values occur where the level curves of f(x,y) are exactly parallel to the constraint g(x,y)=a.

The max and min values occur where the level curves of f(x,y) are exactly perpendicular to the constraint g(x,y)=a.

The max and min values occur where the level curves of f(x,y) the constraint g(x,y) are both equal to 0.

The max and min values have no relationship with the graph of the level curves of f(x,y).

Question 2

Which of the following are true statements of the Lagrange multiplier method? (2 options are correct).

Question 2 options:

The Lagrange Multiplier method is used to optimize (find the max and min values) of a function f(x,y) subject to a constraint g(x,y)=a.

The Lagrange Multiplier method is used to find tangent planes for a 2-variable function.

The Lagrange Multiplier method is computed by setting the gradient of f(x,y) equal to a scalar multiple of the gradient of g(x,y).

The Lagrange Multiplier method is computed by setting the function f(x,y) equal to the function g(x,y).

Question 3

Consider yourself in the middle of solving an optimization problem using the Lagrange multiplier method. Suppose x, y, L are variables and <2x,2y> and <1,2y> are vectors. Suppose also that <2x, 2y> = L <1, 2y> and x^2+y^2=4. What is one way to solve for x and y?

Question 3 options:

Solve the system of equations with three variables and three unknowns:

2x=L

2y=L2y

x^2+y^2=4

Set L=0 and solve for x and y.

Find the magnitude of each of the vectors and set them equal to one another.

Take the cross product of the vectors and set equal to x^2+y^2=4

Question 4

Two techniques for solving maximization with constraint problems include (1) The Lagrange Multiplier method and (2) Plugging in the constraint (by solving for a variable), finding critical points, and evaluating those critical points.

Question 4 options:

Question 5

Suppose a function is given by f(x,y) and a constraint is given by g(x,y)=a. One method of finding the maximum and minimum values of f(x,y) subject to the constraint WITHOUT using Lagrange multipliers might include:

Question 5 options:

Setting the gradient of f(x,y)=0 and solving for x and y.

Setting g(x,y)=0 and f(x,y)=0 and solving for x and y.

Using the constraint function to solve for one variable, plug into f(x,y) and optimize the resulting f function.

Set f(x,y)=g(x,y) and solve for x and y.

The max and min values occur where the level curves of f(x,y) are exactly parallel to the constraint g(x,y)=a.

The max and min values occur where the level curves of f(x,y) are exactly perpendicular to the constraint g(x,y)=a.

The max and min values occur where the level curves of f(x,y) the constraint g(x,y) are both equal to 0.

The max and min values have no relationship with the graph of the level curves of f(x,y).

Explanation / Answer

1) The max and min values occur where the level curves of f(x,y) are exactly parallel to the constraint g(x,y)=a.

2) The Lagrange Multiplier method is used to optimize (find the max and min values) of a function f(x,y) subject to a constraint g(x,y)=a.

3)Find the magnitude of each of the vectors and set them equal to one another

4) true

5)Using the constraint function to solve for one variable, plug into f(x,y) and optimize the resulting f function.

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