Question

A manufacturer 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?

Answer 1

The extrema of a function happen when the derivative is equal to zero. So first we calculate the derivative:
\[R \prime(x) = 100 − 0.4x\]
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:
\[R(250) = 100(250) − 0.2(250^{2})\]
R(250)=$12500