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