Lagrange Multiplier Sample Problem

(Note: highlighted text with the same color represents the same (part) or equal
expression.)

1. Find the maximum and minimum value of f ( x , y )=xy subject to
x +2 y =8.

Solution:
Since the given constraints is x +2 y =8, the constraint function is given
by: g(x , y)=x +2 y .
Using the principle of Lagrange multiplier, the following equation must
be satisfied:
∇ f ( x , y )= λ ∇ g( x , y)
Since ∇ f ( x , y )=¿ f x ( x , y ) , f y ( x , y)>¿ and ∇ g ( x , y )=¿ g x ( x , y ) , g y ( x , y )>¿ , it follows:
¿ f x ( x , y ) , f y ( x , y ) >¿λ¿ g x ( x , y ) , g y (x , y )>¿
By partially differentiating both sides of f ( x , y )=xy with respect to x , we
∂ ∂
got: f x ( x , y )= ∂ x ( xy ). Since ∂ x ( xy )= y , it follows f x ( x , y )= y .
By partially differentiating both sides of f ( x , y )=xy with respect to y , we
∂ ∂
got: f y ( x , y )= ∂ y (xy ). Since ∂ y ( xy )=x , it follows f y ( x , y )=x .
By partially differentiating both sides of g ( x , y )=x +2 y with respect to x ,
∂ ∂
we got: g x ( x , y ) = ∂ x ( x+2 y ). Since ∂ x ( x +2 y )=1, it follows g x ( x , y ) =1.
By partially differentiating both sides of g ( x , y )=x +2 y with respect to y ,
∂ ∂
we got: g y ( x , y )= ∂ y ( x+2 y ). Since ∂ y ( x+ 2 y )=2, it follows g y ( x , y )=2.
Putting it all together, we have:
¿ y , x >¿ λ¿ 1,2>¿
Using the scalar multiple property, we got:
¿ y , x ≥¿ λ , 2 λ >¿
For two vectors to be equal, their corresponding component musts be
equal:

{x=2
y=λ
λ
Substitute x=2 λ and y= λ into the given constraint x +2 y =8, we have:
2 λ+2 ( λ ) =8 → 4 λ=8 → λ=2


Since λ=2, {x=2
y=λ
λ
become { y=2
x=2 ( 2 ) {
→ y =2 . Therefore, the maximum or
x=4
minimum value is at ( 4,2 ) .
Substitute x=4 and y=2 into the given function f ( x , y )=xy , we got
f (4,2)=4( 2)=8.
To determine whether 8 is the maximum or the minimum value is, find
another order pair that satisfy the given constraint.
Notice x=6 and y=1 satisfy the constraint x +2 y =8.