Cost of Manufacturing X Items (average cost minimization)

[theclock54]

Homework Statement

If the cost of manufacturing x items is:
C(x) = (x^3)+21(x^2)+110x+20

Homework Equations

All right, so the first few questions asked for total cost of producing 100 items, and marginal cost. I understood those well. Then it asked for the average cost function, which I found to be C(x)/x. I have a problem where it asks "The production level when the average cost is minimized."

The Attempt at a Solution

Well, I would take the derivative of the average cost function (C(x)/x)
which would give me (2(x^3)+21(x^2)-20)/(x^2)

So then we set that equal to zero, I'm stuck at setting the top equal to zero. How can I factor it? I graphed the average cost function and found x to be approximately -10.40768.
Is there any way to factor it? Or do I have to use a calculator?


Thank you in advance for your replies.

 

[Ray Vickson]

Homework Statement

If the cost of manufacturing x items is:
C(x) = (x^3)+21(x^2)+110x+20

Homework Equations

All right, so the first few questions asked for total cost of producing 100 items, and marginal cost. I understood those well. Then it asked for the average cost function, which I found to be C(x)/x. I have a problem where it asks "The production level when the average cost is minimized."

The Attempt at a Solution

Well, I would take the derivative of the average cost function (C(x)/x)
which would give me (2(x^3)+21(x^2)-20)/(x^2)

So then we set that equal to zero, I'm stuck at setting the top equal to zero. How can I factor it? I graphed the average cost function and found x to be approximately -10.40768.
Is there any way to factor it? Or do I have to use a calculator?


Thank you in advance for your replies.

You get a cubic equation having two negative roots and one positive root. There are formulas available for solving cubic equations, but they are rarely used. A numerical method is preferable in this case. Only positive roots make sense in this problem: we cannot have a negative production level!

RGV