# Total production function using Lagrange Multipliers

1. Mar 1, 2012

### s3a

1. The problem statement, all variables and given/known data
Attached as Question.jpg.

2. Relevant equations
Partial differentiation.
Lagrange multiplier equation.

3. The attempt at a solution
Attached as MyWork.jpg.

Is my work correct? I'm still not confident with myself for these problems and it would be great if someone could confirm if I did this problem correctly and also, the values (or value since I can ignore the negative because it makes no sense in the problem) I get seem rather strange and, knowing my teacher, he usually gives problems with numbers that work out well and his notes also have nicely fitting numbers.

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2. Mar 1, 2012

### Ray Vickson

You made a blunder: you started with
$$L = 12 x^{3/4}\, y^{1/4} + \lambda (100 x + 180 y - 25200)$$ and wrote
$$\partial L/\partial y = 12 x^{3/4}\, y^{-1/4} + 180 \lambda,$$
which is incorrect.

RGV

3. Mar 1, 2012

### s3a

Actually, I don't think so. (If that sounds rude, I don't intend to.)

Looking back at my work here, I see a coefficient of 3 and not 12 for L_y.

4. Mar 1, 2012

### Ray Vickson

You are wrong. Go back and look at what you wrote in the third line of your attachment.

RGV

5. Mar 1, 2012

### s3a

I marked it in red to show what I am talking about in case we're mis-communicating. (It seems to be the third line too.)

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6. Mar 1, 2012

### Ray Vickson

I was not referring to the "3"; I was referring to the $y^{-1/4}.$

RGV

7. Mar 1, 2012

### HallsofIvy

Staff Emeritus
Yes, you have "3" correctly. However, your power of y is wrong. 1/4- 1 is not equal to -1/4.

8. Mar 1, 2012

### s3a

Oh, yeah, oops. :shy:

Is the current attached work correct?

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