First we have to find the profit function P(x).
[tex]\begin{gathered} P\mleft(x\mright)=R\mleft(x\mright)-C\mleft(x\mright) \\ P(x)=41x-2x^2-(x^2+17x+4) \\ P(x)=41x-2x^2-x^2-17x-4 \\ P(x)=-3x^2+24x-4 \end{gathered}[/tex]
The maximum value of the quadratic function is the vertex as it opens downward. Finding the vertex, we have:
[tex]\begin{gathered} Vx=\frac{-b}{2a}=\frac{-24}{2(-3)}=\frac{-24}{-6}=4 \\ \text{ The x-coordinate of the vertex is x=4} \end{gathered}[/tex]
We see that the number of units needed to maximize the profit is 4 units and it satisfies the condition of being between 0 and 13 units.
The answer is x=4.
