Chapter 4 - Applications of the Derivative - 4.1 Maxima and Minima - 4.1 Exercises - Page 243: 53

(a) $n=50$ (b) $45$ people

(a). Note that $P(n) = 50n−.5n^2 −100$, so $P'(n) = 50−n$, which is zero when $n = 50$. It is clear that this is a maximum, since the graph of $P$ is an inverted parabola. (b). Given a domain of $[0, 45]$, since the only critical point is not in the domain, the maximum must occur at an endpoint. Since $P(0) = −100$, and $P(45) = \$1137.50$, he should take $45$ people on the tour.