Lagrange Multipliers - Illinois Institute of Technology
[Pages:18]Lagrange Multipliers
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In This Presentation..
?We will give a definition ?Discuss some of the lagrange multipliers ?Learn how to use it ?Do example problems
Definition
Lagrange method is used for maximizing or minimizing a general function f(x,y,z) subject to a constraint (or side condition) of the form g(x,y,z) =k. Assumptions made: the extreme values exist
g0 Then there is a number such that
f(x0,y0,z0) = g(x0,y0,z0) and is called the Lagrange multiplier.
....
? Finding all values of x,y,z and such that
f(x,y,z) = g(x,y,z)
and
g(x,y,z) =k
And then evaluating f at all the points, the values obtained are studied. The largest of these values is the maximum value of f; the smallest is the minimum value of f.
......
? Writing the vector equation f= g in terms of its components, give
fx= gx
fy= gy fz= gz g(x,y,z) =k
? It is a system of four equations in the four unknowns, however it is not necessary to find explicit values for .
? A similar analysis is used for functions of two variables.
Examples
? Example 1:
A rectangular box without a lid is to be made from 12 m2 of cardboard. Find the maximum volume of such a box.
? Solution:
let x,y and z are the length, width and height, respectively, of the box in meters.
and
V= xyz
Constraint: g(x, y, z)= 2xz+ 2yz+ xy=12
Using Lagrange multipliers,
Vx= gx
Vy= gy Vz= gz 2xz+ 2yz+ xy=12
which become
Continued..
? yz= (2z+y)
(1)
? xz= (2z+x)
(2)
? xy= (2x+2y)
(3)
? 2xz+ 2yz+ xy=12
(4)
? Solving these equations;
? Let's multiply (2) by x, (3) by y and (4) by z, making the left hand sides identical.
? Therefore,
? x yz= (2xz+xy)
(6)
? x yz= (2yz+xy)
(7)
? x yz= (2xz+2yz)
(8)
continued
? It is observed that 0 therefore from (6) and (7) 2xz+xy=2yz+xy
which gives xz = yz. But z 0, so x = y. From (7) and (8) we have 2yz+xy=2xz+2yz
which gives 2xz = xy and so (since x 0) y=2z. If we now put x=y=2z in (5), we get
4z2+4z2+4z2=12 Since x, y, and z are all positive, we therefore have z=1 and so x=2 and y = 2.
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