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:
[tex]\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}(-2x^2+105x-773) \\ =-4x+105 \end{gathered}[/tex]
now we equate it to zero and solve for x:
[tex]\begin{gathered} -4x+105=0 \\ 4x=105 \\ x=\frac{105}{4} \end{gathered}[/tex]
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:
[tex]\begin{gathered} \frac{d^2y}{dx^2}=\frac{d}{dx}(-4x+105) \\ =-4 \end{gathered}[/tex]
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:
[tex]\begin{gathered} y=-2(\frac{105}{4})^2+105(\frac{105}{4})-773 \\ y=605.125 \end{gathered}[/tex]
Therefore the maximum profit is $605
