The cost of an airplane ticket is \$2000 for the first 50 passengers. For the subsequent passengers, a cumulative discount of \$10 is applied. For example, passenger 51 pays \$1990, passenger 52 pays \$1980, and passenger 53 pays \$1970. What is the function that describes the airline's profits? Furthermore, determine the value of x for which the airline's profit is maximized.
My idea is to define the following function that describes the cost of each ticket for passenger number x,
$ f(x) = \begin{cases} 0, \ x=0 \\ 2000, \ 0 \leq x \leq 50 \\ 2500-10x, \ x \geq 50 \end{cases} $
Then the function that describes the airline's profit is
$ R(x) = \int_0^x f(t) dt $
$ R'(x) = -10x+2500 $
The critical point is located at
$ x = 250 $
And this point is a maximum because $R''(x)<0$.
Is this problem correctly solved?
Presuming that you are here to learn , which involves knowing where you are going wrong , I am going to give you "honest" feedback on the Question.
The Question is well written & is using good mathematical notation & is quite clear.
Other that that , it is not at all valid , which is the unfortunate truth . . . .
No matter , it is a learning Opportunity for OP.
Here are the main Points , which are all important to various levels.
(1) Customers are Discrete , not continuous.
(1A) In general , we can not have $0.5$ Customers or $\pi$ Customers.
(1B) It is hence a Discrete Optimization Problem , which might not involve Continuous integrals.
(2) We can not keep Discounting without limit. We should have upper limit on $x$ here.
(2A) When we put $x=1000$ in your function , we get $f(1000)=-7500$ , which means airlines has to pay back ????
(2B) Can the Plane hold millions of Customers ????
(3) The given function is not valid , having Duplicate Definition for at-least 2 $x$ values.
You should make the Intervals non-overlapping. You should also indicate the Domain for $x$ , whether it is Integer & Positive & ETC.
(3A) What is $f(0)$ ? With first line we have $f(0)=0$ , while with second line we have $f(0)=2000$
(3B) Likewise , there is overlap @ $f(50)$ , though luckily both are Same value.
(4) You are calculating revenue only , which can not be Profit. Yo have to involve Costs & then Profit will be given by "Profit = revenue - Costs" . . . .
We can then Optimize that.
Currently , there is no Data on Costs , hence , the Problem is not valid.
(5) When you resolve those Issues , you should try to use Discrete Summation $\Sigma$ to formulate the Problem. That might or might not get converted to Continuous Integral $\int$ , though that is a later concern.
(6) Alternately , you should look into Integer Programming Models , linear Programming & ETC.
It might be much easier to resolve , rather than with Integration.