Paying points for bond rates. Your grandmother wants to buy you a 1-year US Savings Bond. It costs $1,000. It earns interest at a continuous rate of r% for a year, then you get to redeem it for its final value. Your grandmother gets to choose r, but each percentage point costs $8, deducted from the face value of the bond at the start. In each part of this problem, please name functions explicitly and state what optimization problem you are solving each time you maximize something.

 (a) How many points should your grandmother buy, in order to optimize the value at the end of a year? 

(b) Suppose the amount of $8 they charge per point is replaced by some other value, L. How large can L be before it is no longer worth buying any points? What does that mean in terms of the maximization problem you solved in part (a)? 

(c) How sensitive is your financial gain on the $1,000 bond to L? Specifically what is your marginal rate of increased value per increase in L when L = 8? 3