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A family has $100,000 in savings accounts. They seek financial advice to develop a ten-year housing strategy. The two options are: continue renting the apartment or take a bank loan to buy a property. (a) Assume that they spend $19.000 per annum on renting the apartment and put their $100,000 savings in a bank deposit at the interest rate 2% per annum compounded annually. Find the family 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 the family takes a bank loan of $400,000 for the period of ten years to buy a property worth $500,000. This is an interest only loan at 5% per annum so the total sum to be repaid to the bank after ten years is $600,000 (one and a half of the original amount). 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 frent (t) and four(t), where t = [0,10). Is frent - fbuy convex or concave function? Decreasing or increasing? (d) Compare frent (10) and fbuy (10). (e) What would be the answer in (d) if the property value does not grow over the years? What if it grows at a moderate 1% per annum, corresponding to the population growth rate

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