A formula proposed by Hurley for the red cell volume (RCV) in milliliters for males is RCV = 1486S² - 4106S + 4514, where S is the surface area (in square meters). A formula given by Pearson et al. is RCV = 1486S - 825.

(a) For the value of S for which the RCV values given by the two formulas are closest, find the rate of change of RCV with respect to S for both formulas. What does this number represent?

(b) The formula for plasma volume for males given by Hurley is PV = 995e^0.6085S, while the formula given by Pearson et al. is PV = 1578S, where PV is measured in milliliters and S in square meters. Find the value of S for which the PV values given by the two formulas are the closest. Then find the value of PV that each formula gives for this value of S.

(c) For the value of S found in part b, find the rate of change of PV with respect to S for both formulas. What does this number represent?

(d) Notice in parts (a) and (c) that both formulas give the same instantaneous rate of change at the value of S for which the function values are closest. Prove that if two functions f and g are differentiable and never cross but are closest together when x = x₀, then f'(x₀) = g'(x₀).