Question: If you invest x dollars in a certain project, your return is R(x). You want to choose x to maximize your return per dollar invested,8 which is
r(x) = R(x)/x
(a) The graph of R(x) is in Figure 4.80, with R(0) = 0. Illustrate on the graph that the maximum value of r(x) is reached at a point at which the line from the origin to the point is tangent to the graph of R(x).
(b) Also, the maximum of r(x) occurs at a point at which the slope of the graph of r(x) is zero. On the same axes as part (a), sketch r(x). Illustrate that the maximum of r(x) occurs where its slope is 0.
(c) Show, by taking the derivative of the formula for r(x), that the conditions in part (a) and (b) are equivalent: the x-value at which the line from the origin is tangent to the graph of R is the same as the x-value at which the graph of r has zero slope.
