Chapter 6.1, Problem 50E

EXERCISES

Satisfaction Suppose some substance (such as preferred food) gives satisfaction to an individual, but the substance requires effort to obtain, so that after a while the individual is no longer interested in expending more effort to obtain the substance. A mathematical model of this situation is given by

$S=a\ln kx-bx,$

Where $S$ is the amount of satisfaction. $x$ is the amount of the substance, and $a,b$ and $k$ are constants. Source: Mathematical Biology of Social Behavior. Find the amount of the substance that gives the maximum amount of satisfaction.