Question

Diminishing Returns. Spending money on advertising for a product can increase the amount of revenue generated by selling that product. Eventually, however, the amount by which revenue increases is offset by the cost of the advertising. The revenue generated (in thousands of dollars) by spending x thousands of dollars advertising a certain product is measured and found to be

R(x) = x3e−0.3x,where x is at most 10 (that is, $10,000).

6. If R′(x) is negative, it means that spending more money will actually reduce sales. IsR′(x) ever negative on the interval 0 ≤ x ≤ 10?

7. We would like to find the point at which R′(x) stops increasing. This is called the point of diminishing returns.

   (a) Calculate R′′(x).
   (b) On what intervals is R′(x) increasing and on what intervals is R′(x) decreasing?

   (c) What is the point of diminishing returns for this particular product?

Answer 1

(a)We have:

(b) At , we have .

goes from increasing to decreasing and vice versa whenever changes sign, i.e. has a root. In the interval ,   has roots, one at approximately , the other at (found by solving and taking the values . The third root is approximately at ).

Thus, is decreasing in , then increasing in and again decreasing in .

Also note the graph:

(c) stops increasing from onwards. This is the point of diminishing returns for this particular product.