2nd Derivative's Relevance to Cost and Revenue Functions

[kgeorge1]

Hi there,

This isn't a procedure problem, what I am having difficulties with is how the second derivative effects cost and revenue functions. I believe that it has something to do with how the second derivative allows you to find the abs. max and abs. min of a function, but part of my is saying that that is way off base. Or another thought that came to me is possibly optimization, but I'm skeptical about that.

So my question: What does the second derivative reveal about cost and revenue functions?

I'm not looking for a lecture, or a perfect answer, just a couple of examples perhaps. Any help I could get would be appreciated.

Thanks!

 

[galactus]

The 'law of diminishing returns' is related to the second derivative.

Example:

Th revenue R(x) generated from sales of a certain product is related to the amount x spent on advertising by

\(\displaystyle R(x)=\frac{1}{150000}(600x^{2}-x^{3}), \;\ 0\leq{x}\leq{600}\), where x and R(x) are in thousands.

Since a point of diminishing returns occurs at inflection points, look for an x value that makes R''(x)=0.

\(\displaystyle R''(x)=\frac{1}{125}-\frac{1}{25000}x\)

Set to 0 and solve for x, we get x=200.

Test a number in the interval (0,200) to see that R''(x) is positive there. Then test it in (200,600) and see it is negative there. Since R''(x) changes from positive to negative at x=200, the graph changes concavity and there is a point of diminsihing returns at the inflection point \(\displaystyle (200,\frac{320}{3})\)

Any advertisement beyond $200,000 would not pay off

Does that example help?.

 

[kgeorge1]

Yes, it helps greatly. It certainly gives me a basis to go off of. Thank you for your help!

Best Regards-