1 Answer

Roy Reznik · Answer 1 · 2023-08-02T01:16:11+0000

To find the maximum profit we need to maximize the function.

First we need to find the critical points, to do this we need to find the derivative of the function:

$\begin{gathered} (dy)/(dx)=(d)/(dx)(-2x^2+105x-773) \\ =-4x+105 \end{gathered}$

now we equate it to zero and solve for x:

$\begin{gathered} -4x+105=0 \\ 4x=105 \\ x=(105)/(4) \end{gathered}$

hence the critical point of the function is x=105/4.

The next step is to determine if the critical point is a maximum or a minimum, to do this we find the second derivative:

$\begin{gathered} (d^2y)/(dx^2)=(d)/(dx)(-4x+105) \\ =-4 \end{gathered}$

Since the second derivative is negative for all values of x (and specially for x=105/4) we conclude that the critical point is a maximum.

Hence the function has a maximum at x=105/4. To find the value of the maximum we plug the value of x to find y:

$\begin{gathered} y=-2((105)/(4))^2+105((105)/(4))-773 \\ y=605.125 \end{gathered}$

Therefore the maximum profit is $605

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