Section 3.9 Lagrange Multipliers

3.9 Lagrange Multipliers

In previous section, we solve optimization problems using second derivative test or the closed boundary method using two variable functions. Those method may not work so well if one can not reduce the problem into a two variable problem. We introduce the new method, Lagrange multiplier method to solve optimization problems with constraints.

Theorem: Method of Lagrange Multipliers: One Constraint 

Let $f$ and $g$ be functions of two variables (three variables) with continuous partial derivatives at every point of some open set containing the smooth curve $g(x,y)=k$ (smooth surface $g(x,y,z)=k$). Suppose that $f$ , when restricted to points on the curve $g(x,y)=k$ ( $g(x,y,z)=k$), has a local extremum at the point $(x_0,y_0)$ ($(x_0,y_0,z_0)$)and that $\mathrm{\nabla }g(x_0,y_0)\ne 0$ ($\mathrm{\nabla }g(x_0,y_0,z_0)\ne 0$). Then there is a number $\lambda$ called a Lagrange multiplier, for which $\mathrm{\nabla }f(x_0,y_0)=\lambda \mathrm{\nabla }g(x_0,y_0)$ ($\mathrm{\nabla }f(x_0,y_0,z_0)=\lambda \mathrm{\nabla }g(x_0,y_0,z_0)$). 

Problem-Solving Strategy: Steps for Using Lagrange Multipliers 

1. Determine the objective function $f(x,y)$ ($f(x,y,z)$) and the constraint function $g(x,y)$ ($g(x,y,z)$). Does the optimization problem involve maximizing or minimizing the objective function? 

2. Set up a system of equations using the following template: $\mathrm{\nabla }f(x,y)=\lambda \mathrm{\nabla }g(x,y)$ and $g(x,y)=k$. ($\mathrm{\nabla }f(x,y,z)=\lambda \mathrm{\nabla }g(x,y,z)$ and $g(x,y,z)=k$).

3. Solve for $x$ and $y$ ($z$). 

4. The largest of the values of $f$ at the solutions found in step 3 maximizes $f$ ; the smallest of those values minimizes $f$.

Example 1: Use the method of Lagrange multipliers to find extremum values of $f(x,y)=x^2+y^2-2x+4y$ subject to the constraint $x+2y=4$.

Exercise 1: Use the method of Lagrange multipliers to find extremum values of $f(x,y)=x^2+y^2+4x-6y$ subject to the constraint $2x+y=6$.

Example 2: Find extremum values of the function $f(x,y,z)=x^2+y^2+z^2$ subject to the constraint $x+y+z=4$.

Exercise 2: Find extremum values of the function $f(x,y,z)=x^2+y^2+z^2$ subject to the constraint $x+y+z=1.$

Example 3: Use the method of Lagrange multipliers to find the point on the plane $2x+3y+z=6$ that is closest to the point $(0,1,-1).$

Exercise 3: Use the method of Lagrange multipliers to find the point on the plane $x+2y+z=4$ that is closest to the point $(1,-1,0).$

Example 4: Use the method of Lagrange multipliers to find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the plane $x+2y+3z=6.$

Exercise 4: Use the method of Lagrange multipliers to find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the plane $x+y+z=1.$

Group work:

1. Use the method of Lagrange multipliers to find extremum values of the function $f(x,y,z)=x+3y-z$ subject to the constraint $x^2+y^2+z^2=1$.

2. A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed $96$ in. Use the method of Lagrange multipliers to find the dimensions of the box that meets this condition and has the largest volume.

3. Use the method of Lagrange multipliers to find the maximum volume of a cylindrical soda can such that the sum of its height and circumference is $120$cm.