S represents weekly sales of a product. What can be said of S' and S'' for each of the following?
(a) the rate of change of sales is increasing
(b) sales are increasing at a slower rate
(c) the rate of change of sales is constant
(d) sales are steady
(e) sales are declining, but at a slower rate
(f) sales have bottomed out and have started to rise
First derivative S':
1. If S' is positive then S is increasing
2. If S' is negative then S is decreasing
3. If S'=0 then S has an extreme (maximum or minimum value)
Second derivative S'': If there exist an x value such that S'(x)=0 then the second derivative determines whether x is a maximum, a minimum, or neither.
The solution provides detailed explanations on the relationship of function and its derivatives. It also provides full answer for the problem.