Lagrange Multipliers. Joseph Louis Lagrange was born in Turin, Italy in Beginning

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1 Andrew Roberts 5/4/2017 Honors Contract Lagrange Multipliers Joseph Louis Lagrange was born in Turin, Italy in Beginning at age 16, Lagrange studied mathematics and was hired as a professor by age 19. By age 20, Lagrange sent Euler improved solutions for deriving the central equation in the calculus of variations. By age 25, Lagrange was already considered one of the greatest living mathematicians (Seikali). In 1776, Lagrange was recommended to succeed Euler as the director of the Berlin Academy by Euler himself (Seikali). Constantly being offered new honours, accolades, and positions in many countries and universities, Lagrange spent his life in various parts of Europe teaching or contributing to the field. During his height as a professor in 1797 at Ecole Polytechnique, it was said that he taught so thoroughly and effectively that his students would, almost unintentionally, contribute to the extension of the subject of mathematics (Ball). Throughout his life, Lagrange contributed to almost all branches of mathematics, the most important of which were the previously mentioned calculus of variations, as well as the solution to polynomial equation and important contributions to the solutions to power series and functions. Some of these important contributions were: creating a better way to solve Euler s equation, defining volume and surface area using double integrals, and coming up with the

2 notation for surface integrals in general. Additionally, Lagrange began creating what is now known as the fundamental theorem of calculus (Seikali). The subject of this paper, the method of Lagrange multipliers, is a strategy developed by Lagrange to find the extrema of a function with constraints. This is extremely helpful in calculus, because it is an optimization of an otherwise very difficult and tedious method to find maxima and minima on the boundaries of a function within a constraint (Paul s Online Notes). In mathematical terms, Lagrange multipliers are used to find the extrema of a function f(x,y,z) which is subject to the constraint g(x,y,z) = k. The method is really only a two step logical process: 1. First, the following system of equations must be solved ( λ is an unknown constant known as the Lagrange Multiplier ): f( x, y, z) = λ g( x, y, z) g (x, y, z) = k 2. After solving the system, plug in the solutions for x, y, z into the original function f(x,y,z), allowing one to identify the extrema. To solve the system of equations in the first step, it helps to first fully expand the vector equation into its 3 component linear equations, shown below: f( x, y, z) = λ g( x, y, z) < f x, f y, f z > = < λg x, λg y, λg z > f x = λg x f y = λg y f x = λg y g (x, y, z) = k

3 In this case we have 4 variables (x, y, z, and λ ) and 4 equations, so the system can be solved to find a real solution which will help identify the extrema. It is important to note, however, that it could be possible that there are no extrema, so it is necessary to make sure that any answers are possible within the context of the problem (Paul s Online Notes). To understand how this method works in application, an example of optimizing the volume of a box with a constrained surface area will be worked: Suppose that one is given 36 square centimeters of material to make a box of maximal volume. How could the dimensions of this box that allow it to have the most volume with this given surface area be found? Conveniently, the method of Lagrange Multipliers is perfect for optimization problems like this one, so each step of solving this problem using Lagrange Multipliers will be worked through below: 1. First, identify the function f( x,y,z ) to be identified and the constraint g( x,y,z ) = k that the function must follow. The problem asks for largest volume, so the function to be optimized is the volume of a box, f (x, y, z) = x * y * z = xyz The constraint is that the surface area of the box is a total of 36 square centimeters, so, g (x, y, z) = 2 (xy + xz + yz) = 36 g (x, y, z) = xy + xz + yz = Set up the system of equations using the Lagrange Multiplier Remember the following relationship between the gradient of the functions: f( x, y, z) = λ g( x, y, z) < f x, f y, f z > = < λg x, λg y, λg z >

4 f x = λg x f y = λg y f x = λg y Using this method, the four equations that compose this system are the following: y z = λ (y + z) x z = λ (x + z) x y = λ (x + y) 3. Solve the system of equations x y + xz + yz = 18 The system can be solved using any preferred method, but one effective solution is to multiply each of the first three equations by the variable they are missing so that each of them is equal to xyz. This method will be shown below: x yz = x λ(y + z) x yz = y λ (x + z) x yz = z λ (x + y) Now, set the first two equations equal to each other to begin solving x λ(y + z) = y λ(x + z) λ (xy + x z) λ(xy + yz) = 0 λ xy + λx z λxy λyz = 0 λx z λyz = 0 λ(x z yz) = 0 This gives two solutions, λ =0 or xz = yz. It can quickly be determined that λ =0 is not helpful since it would mean that the volume (and thus one or more of the side-lengths) would be 0, which is obviously not possible in the case of a box with real, non-zero dimensions. Therefore, xz = yz, which means that x = y.

5 Performing the same operations on the second and third of the equations above gives: y λ(x + z) = z λ(x + y) λ (xy + y z) λ(xz + yz) = 0 λ xy + λy z λxz λyz = 0 λxy λxz = 0 λ(x y xz) = 0 In this case it is found that xy = xz, or z = y. Therefore, it can be found that, x = y = z Finally, this knowledge can be used to plug into the final equation, solving the system: x y + xz + yz = 18 x 2 + x 2 + x 2 = 18 3x 2 = 18 x = ± 6 However, obviously the physical box in question cannot have a negative dimension, therefore, x = y = z = 6 4. Verify the solution The final step to solving this problem is to ensure that the value of the dimensions obtained (forming a cube of sidelength 6 ), are actually a maxima, maximizing the volume of the box. The method of Lagrange Multipliers gives a set of points (x, y, z) that can either maximize or minimize the function under a given constraint (Paul s Online Notes). In this case, we can verify

6 that these points do form a maximum since we know that if any dimension were to increase, the others would decrease because of the equation for surface area: xy + xz + yz = 32. One interesting development when using Lagrange Multipliers is that the value of λ itself is rarely used, or even found, as could be seen in the example. This is normal, as λ is just a tool used to relate the constraint to the original function in order to create a system of equations. Regardless, as shown in the above example, the method of Lagrange Multipliers is a useful tool to optimize any function with a given restraint, allowing for a much more efficient method than checking each of the boundaries of the function individually for extrema. Lagrange multipliers can even be extended to solve functions in three dimensions with multiple constraints by setting up a system of 5 equations (in the case of two constraints) or more depending on how many constraints are present. Lagrange lived an extraordinary life, and was widely considered one of the greatest living mathematicians during his lifetime. The method of Lagrange Multipliers is just the smallest fraction of his work on the field of calculus alone, not to mention other fields of mathematics, and yet it is a useful and sophisticated method to solve for extrema in constrained functions. Hopefully this discussion of Lagrange Multipliers not only provides information about a helpful mathematical tool in the field of calculus, but also provides insight into the genius of Joseph Louis Lagrange.

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8 Works Cited Ball, W. W. Rouse. A short account of the history of mathematics. 4th ed. New York: Main Street, Print. Seikali, Nahla. "Joseph-Louis Lagrange." Berkeley Math. N.p., n.d. Web. 05 May < "Calculus III - Lagrange Multipliers." Paul's Online Notes. N.p., n.d. Web. 05 May <