Another common use of cubic function is in the advertising industry. The volume of sales of some products, like automobile or beer, can be approximated as the cubic function of the amount of advertising. Advertisers refer to the point where the rate of change in sales reaches its maximum (which coincides with the inflection point of the sales function) as the point of diminishing return. Beyond this point the contribution of additional advertising to sales declines.
Consider the following function relating number of units (in 100) of a product sold, N , to the amount (in $1,000) spend on advertising, S.
N = -0.08S3 + 3.6S2 - 30S + 80 , 5 ≤ S ≤ 25
Find the function's point of diminishing return.