I have no idea how to do this, I tried a lot of things but they don't make sense and I have too many variables.
A manufacturer has been selling lamps at the price of \$6/lamp, and at this price they have been selling 3000 lamps a month. The manufacturer wishes to raise the price and estimated that for each \$1 increase they will sell 1000 fewer lamps a month. The manufacturer can produce the lamps at a cost of \$4 per lamp Express the manufacturers monthly profit as a function of the price that the lamps are sold, draw the graph and estimate the optimal selling point.
I think the profit should be $\#(\mathrm{lamps\ sold})\cdot(\mathrm{price\ of\ lamps}) - 4\cdot\#(\mathrm{lamps\ sold})$.
If the manufacturer decides to set her price to \$$p$ then her estimation of the number of sold lamps per month is $$ 3000-(p-6)1000=9000-1000p. $$ Hence the monthly profit is $$ (9000-1000p)(p-4)=-1000p^2+13000p-36000. $$ Computing the derivative and setting it to zero shows that the optimal selling point is $p^*=6.5$.