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

Find the number of units of grain that are to be produced to maximize the profit if-example-1

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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

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