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(Note: highlighted text with the same color represents the same (part) or equal
expression
...
Find the maximum and minimum value of f ( x , y ) =xy subject to
x+ 2 y =8
...

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 ¿ gx ( 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 )
...

By partially differentiating both sides of f ( x , y ) =xy with respect to y
∂
∂
, we got: f y ( x , y ) = ∂ y ( xy )
...

By partially differentiating both sides of g ( x , y )=x +2 y with respect to
∂
∂
( x +2 y )=1 , it follows
x , we got: gx ( x , y )= ( x +2 y)
...


By partially differentiating both sides of
y , we got:

∂
(x +2 y )
...


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:

{xy=λ
=2 λ
Substitute
have:

x=2 λ and

y=λ

into the given constraint

x+ 2 y =8 , we

2 λ+2 ( λ )=8→ 4 λ=8 → λ=2
y=2
x =2 ( 2 )
maximum or minimum value is at ( 4,2 )
...
Therefore, the

Substitute

x=4
f (4,2)=4(2)=8
...

Notice x=6 and y=1 satisfy the constraint x+ 2 y =8
...
There is no
relative minimum

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