Amanufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function r(x) = 100x − 0.2x2, where the revenue r(x) is measured in dollars. what is the maximum revenue, and how many units should be manufactured to obtain this maximum?

The extrema of a function happen when the derivative is equal to zero. So first we calculate the derivative:R′(x)=100−0.4x
And then we solve:0=100−0.4x
0.4x=100
x=250 units
Now we can substitute back into the original equation to calculate the revenue:R(250)=100(250)−0.2(2502)
R(250)=$12500