Michael Corral: Vector Calculus

96 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES
2.7 Constrained Optimization: Lagrange Multipliers
In Sections 2.5 and 2.6 we were concerned with finding maxima and minima of functions
without any constraints on the variables (other than being in the domain of the
function). Whatwouldwedoiftherewereconstraintsonthevariables? Thefollowing
example illustrates a simple case of this type of problem.
Example 2.24. Forarectanglewhoseperimeteris20m, findthedimensionsthatwill
maximize the area.
Solution: The area A of a rectangle with width x and height y is A= xy. The perimeter
P of the rectangle is then given by the formula P=2x+2y. Since we are given that the
perimeter P=20, this problem can be stated as:
Maximize : f(x,y)= xy
given : 2x+2y=20
The reader is probably familiar with a simple method, using single-variable calculus,
for solving this problem. Since we must have 2x+2y=20, then we can solve for, say,
y in terms of x using that equation. This gives y=10− x, which we then substitute
into f to get f(x,y)= xy= x(10− x)=10x− x 2 . This is now a function of x alone, so we
now just have to maximize the function f(x)=10x− x 2 on the interval [0,10]. Since
f ′ (x)=10−2x=0⇒ x=5and f ′′ (5)=−2