Let V(t) be the value of an asset t years from now, and assume that the prevailing annual interest rate remains fixed at r (expressed as a decimal) compounded continuously.

a. Show that the present value of the asset P(t) = V(t)e^−rt has a critical number where V'(t) = V(t)r. (Using economic arguments, it can be shown that the critical number corresponds to a maximum.)

b. Explain why the present value of V(t) is maximized at a value of t where the percentage rate of change (expressed in decimal form) equals r.