Question

Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartments is given by the following function.

1.) To maximize the monthly rental profit, how many units should be rented out?
______ units

2.) What is the maximum monthly profit realizable?  
$ _______

P(x) = x^3 + 3x^2 +3 on [-3,2]

Answer 1

we have,

we have to find the values of x such that profit is maximum given that x must be inside the interval [-3,2]

we can write,

equate it to 0 we can say that,

Hence we can say that x = 0 and x = -2 are two critical point of the profit function which is inside [-3, 2]

Ideally to find the maximum value of P(x) on the interval [-3,2] we have to evaluate P(x) at x = -3, x = -2, x = 0 and x = 2

But we know that units of apartment rented out can not be negative hence evaluate P(x) at x = 0 and x = 2 to find the maximum profit

we have,

Hence,

we can see that maximum profit is P(2) = $23

Hence we can write,

To maximize the monthly rental profit x = 2 units should be rented out

maximum monthly profit realizable is P(2) = $23