1. Assume you are a family who has $100,000 in Commonwealth Bank account. You have two housing options: to take a bank loan and buy a property or continue to rent an apartment.

(a) Assume that you spend $19,000 per year on renting the apartment and put $100,000 [2] savings in a bank deposit at the interest rate 2% per year compounded annually. Find your net worth (calculated as the money on the deposit minus the rent paid, in thousand dollars) after t years. Define the function frent : [0, 10] R by allowing t to be a real number in the expression for the net worth.

(b) Now assume that you can takes a bank loan of $400,000 for the period of ten years to [3] buy a property worth $500,000. The total sum to be repaid to the bank after ten years is $650,000. Suppose the loan is paid regularly so it is described by a linear function, whose value at t = 10 equals the total sum to be repaid to the bank. The property value

is expected to grow at the rate 2% per annum. Similarly to (a), define the function fbuy : [0,10] R corresponding to the family net worth (calculated as the property value minus the loan paid, in thousand dollars) after t years.

(c) Compute the derivatives f (t) and f (t), where t [0, 10]. Is f rent buy

concave function? Decreasing or increasing?