Expert: Steve Holleran - 4/17/2007

Question
hallo sir, im from a university student in malysia and taking calculus, i have a big problem to factor this equition can you help me? the question is suppose that c(x) = X^3 - 20X^2 + 2000X is the cost of manufacturing items. find a production level that will minimize the average cost of making items.. this differentiation chapter.. as soon as possible.. thank you.

Answer
Hello eein,

Okay, let's see here, if the function you give,

C(x) = x^3 - 20x^2 + 2000x is the cost of making x items, then the average cost would be this expression, the total cost, divided by x, the number of items:

A(x) = C(x) / x = x^2 - 20x + 2000.

Now to find the critical points, we need to take the derivative:

A'(x) = 2x - 20 and set this = 0:

A'(x) = 2x - 20 = 0 ------> x = 10 is the critical value.

Now, using the Second Derivative Test to determine the type of critical value we just obtained,

A''(x) = 2 .  Since A'' is positive, the critical value is a minimum for the function.

So,  a production level of x = 10 items will minimize the average cost for this situation.

I hope I have interpreted your question correctly, and that this explanation is clear.  Good luck.

Steve Holleran