When interest rates are low, many homeowners take the opportunity to refinance their mortgages. As rates start to rise, there is often a flurry of activity as latecomers rush in to refinance while they still can do so profitably. Eventually, however, rates reach a level where refinancing begins to wane. Suppose in a certain community, there will be M(r) thousand refinanced mortgages when the 30-year fixed mortgage rate is r%, where

a. For what values of r is M(r) increasing?

b. For what interest rate r is the number of refinanced mortgages maximized? What is this maximum number?

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M(r) = 1 + 0.05r 1 + 0.0047² for 1 ≤ r ≤ 8