Maximizing profit ?Suppos ? e a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for t?aking n? people on a city ?to?ur is? ? ) = ?? 0 ? 0.5?n)?100. (Al? hough ?P is defined only for positive integers, treat it as a continuous function.) a. How many people should the guide take on a tour to maximize the profit? b. Suppose the bus holds a maximum of 45 people. How many people should be taken on a tour to maximize the profit?

STEP_BY_STEP SOLUTION Step-1 Let f be a continuous function defined on an open interval containing a 1 number ācā.The number ācā is critical value ( or critical number ). If f (c) = 0 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-2 When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum. Let f be a func tion with domain D and let c be a fixed constant in D . Then the output value f ) is the 1. Absolute maximum value of f on D if and only if f(x) f(c) , for all x in D. 2. Absolute minimum value of f on D if and only if f(c) f(x) , for all x in D. Step_3 a). Given is ; A tour guide has a bus that holds a maximum of 100 people, and also given that tour guide has a profit (in dollars) for taking n people on a city tour is ; P(n) = n( 50 - 0.5n ) - 100 , here P is defined only for positive integers. Clearly the function is polynomial function , so this is continuous on this domain. Now , we have to find the critical points of the revenue function.For that we have to evaluate the function at critical points.so, the critical points satisfies the equation . Given function is; P(n) = n( 50 - 0.5n ) - 100 . For the critical points we have to...