Question

Find the number of units of grain that are to be produced to maximize the profit if…

Answer 1

we need to make revenue-cost and then maximize

`\begin{gathered} R(x)-C(x) \\ (97x-2x^2)-(2x^2+49x+6) \end{gathered}`

simplify

`\begin{gathered} =97x-2x^2-\mleft(2x^2+49x+6\mright) \\ =97x-2x^2-2x^2-49x-6 \\ =-2x^2-2x^2+97x-49x-6 \\ =-4x^2+97x-49x-6 \\ =-4x^2+48x-6 \end{gathered}`

now, to maximize, we need to find the derivate and make it equal to 0

`\begin{gathered} (d)/(dx)(-4x^2+48x-6)=0 \\ -8x+48=0 \\ -8x=-48 \\ (-8x)/(-8)=(-48)/(-8) \\ x=6 \end{gathered}`

so, the maximum profit is at x = 6