Maximization subject equality constraint

[oswald]

Homework Statement

Max f = x²yz³
sub. 50x+10y+100z = 1000

Homework Equations

Using Lagrange:

L = x²yz³ - λ ( 50x + 10 y + 100z - 1000 )
Lx = 2xyz³ - λ50 = 0
Ly = x²z³ - λ10 = 0
Lz = 3x²yz² - λ100 = 0

i found z = 2,5 x= 10 y=25, what's wrong?

The Attempt at a Solution

z = 5
x = 20/3
y = 50/3

 

[Pere Callahan]

Did you check if your solutions satisfy the four equations, (which appear to be right):

Lx = Ly =Lz = 0 and the constraint 50x+10y+100z = 1000 ?

If they do your solution is correct, if not you made a mistake in solving the four equations and you should show us how you tried to do it so that we can help you.

Note, that you have four equations and four variables to solve for: x, y, z and lambda.

 

[oswald]

2xyz³/50 =λ
x²z³/10 =λ
3x²yz²/100= λ
hence,
2xyz³/50 = x²z³/10 = 3x²yz²/100

substituting:

2xyz³/50 = x²z³/10
2y/50 = x /10
20 y = 50x

x²z³/10 = 3x²yz²/100
z/10 = 3y/100
100 z = 30y

50x+10y+100z = 1000
20y + 10y + 30y = 1000
60y = 1000
y = 50/3
ah, i made a mistake in solving the four equations... thanks

 

[Pere Callahan]

It's always a good idea to double-check your calculations if the result appears to be wrong.