Constrained Optimization: Lagrange Multipliers

A cylindrical soda can holds about 355 cc of liquid. In this activity, we want to find the dimensions of such a can that will minimize the surface area, since this will minimize the material used to construct this can. For the sake of simplicity, assume the can is a perfect cylinder.

1. What are the variables in this problem? Based on the context, what restriction(s), if any, are there on these variab…

Use the method of Lagrange multipliers to find the dimensions of the least expensive rectangular packing crate with a volume of 240 cubic feet when the material for the top costs $2 per square foot, the bottom costs $3 per square foot, and the sides cost $1.50 per square foot.

The Cobb-Douglas production function is used in economics to model production levels based on labor and equipment. Suppose we have a specific Cobb-Douglas function of the form where $x$ is the dollar amount spent on labor and $y$ the dollar amount spent on equipment. Use the method of Lagrange multipliers to determine how much should be spent on labor and how much on equipment to maximize productivi…

Use the method of Lagrange multipliers to find the point on the line $x-2y=5$ that is closest to the point $(1,3)$. To do so, respond to the following prompts.

1. Write the function $f=f(x,y)$ that measures the square of the distance from $(x,y)$ to $(1,3)$. (The extrema of this function are the same as the extrema of the distance function, but $f(x,y)$ is simpler to work with.)

2. What is the constraint $g(x,y) = c$?

3. Write the equations resulting from $\nabla f = \lambda \nabla g$ an…

Apply the Method of Lagrange Multipliers to solve each of the following constrained optimization problems.

1. Determine the absolute maximum and absolute minimum values of $f(x,y) = (x-1)^2 + (y-2)^2$ subject to the constraint that $x^2 + y^2 = 16$.

2. Determine the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to and farthest from the point $(3,1,-1)$. (As in the preceding exercise, you may find it simpler to work with the square of the distance formula, rathe…

In this exercise we consider how to apply the Method of Lagrange Multipliers to optimize functions of three variable subject to two constraints. Suppose we want to optimize $f = f(x,y,z)$ subject to the constraints $g(x,y,z) = c$ and $h(x,y,z) = k$. Also suppose that the two level surfaces $g(x,y,z) = c$ and $h(x,y,z) = k$ intersect at a curve $C$. The optimum point $P = (x_0,y_0,z_0)$ will then lie on $C$.

1. Assume that $C$ can be represented parametrically by a vector-valued function $\vr = \vr(t)$…

There is a useful interpretation of the Lagrange multiplier $\lambda$. Assume that we want to optimize a function $f$ with constraint $g(x,y)=c$. Recall that an optimal solution occurs at a point $(x_0, y_0)$ where $\nabla f = \lambda \nabla g$. As the constraint changes, so does the point at which the optimal solution occurs. So we can think of the optimal point as a function of the parameter $c$, that is $x_0 = x_0(c)$ and $y_0=y_0(c)$. The optimal value of $f$ subject to the con…