Answer :
We must find the maximum of the following function:
[tex]P(x)=-4x^2+8x+4.[/tex]To find the maximum, we equal to zero the first derivative of P(x):
[tex]P^{\prime}(x)=-4\cdot2x+8=0,[/tex]and then we solve for x:
[tex]\begin{gathered} -4\cdot2x+8=0, \\ -8x+8=0, \\ 8x=8, \\ x=\frac{8}{8}=1. \end{gathered}[/tex]Now, we evaluate the function for x = 1, we get:
[tex]P(1)=-4\cdot1^2+8\cdot1+4=8.[/tex]From the statement we know that:
• x is the number of units produced per week, in thousands,
,• P(x) is the weekly profit, in hundreds of dollars.
So the maximum is reached for 1,000 units and the profit is $800 in that case.
Answer
b. 1,000 units; $800