What is the number of orders needed to produce maximum profit from the equation `P(x)=-x^2+1250x-271600` ?

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I would like to add to the above answer. One can use basic calculus, to be specific the use of basic differentiation: 

`(dP)/dx = -2x + 1250` (Applying basic differentiation) 

In order to find the maximum make `(dP)/dx =0`

`0 = -2x + 1250`

Now solve for x:

`2x = 1250`

`x = 625`

Now substitute the answer  into the original equation to find the maximum profit : 

`P(x) = - x^2 + 1250x - 271600`

`P(625) = - (625)^2 + 1250 (625) - 271600 = 119 025`


(Calculus is another simple way to solve problems when determining a question asking for the maximum. Only use calculus if you are familiar with it, otherwise use the methods as stated in the previous answer.)

  • When finding the maximum, first differentiate with respect to the independent variable, many times will be x, then make it equal to zero and solve. 
  • Answer: x = 625 maximum profit P(625) = 119 025



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I think `x` is the number of orders and `P` is the profit.

The graph of this function is a parabola branches down. Its maximum is reached at a vertex, whose x-coordinate is `-b/(2a)`  (for the equation `y=ax^2+bx+c`). So here it is `x=-1250/(2*(-1))=1250/2=625.`

If we don't know about parabolas, no problem. Simply recall that the necessary condition for an extremum is `P'(x)=0` and obtain the same answer. If we don't know derivatives, extract a perfect square:


It is always `gt=119025` and is equal to it for `x=625.`


The answer is x=625 and the maximum profit is 119025.

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