A researcher is modelling the evolution of a company's market share in time using the function

f(x) = Rx/(1 + Mx)

Here x represents the company's market share in a particular year, and f(x) the market share in the following year (test your understanding: what would f(f(x)) represent?). The parameters R, M (which depend on the company and its relationship to the market) satisfy M > R-1> 0. Since x represents a proportion of the market, the domain of f is the interval 0, 1).

1. (a) Solve the equation f(x) = x.

(b) Describe in a paragraph what the solution(s) found in part (a) represent, in terms of market shares.

2. (a) What polynomial function approximates f(x) well for small values of x? Explain your reasoning.

(b) In a paragraph, interpret your answer in part (a) in terms of market shares: what does your answer to part (a) tell you about how this model describes companies with small market share? what is the significance of the constant R? (recall the assumption R > 1)

3. (a) It may be shown that f' (x) = 1+ Mays, which is always positive. Describe in a paragraph what the feature f' (x) > 0 for all x > 0 implies, in terms of market shares.

(b) Why might the feature f' (x) > 0 for all > 0 be unsuitable for modelling certain market situations? Describe your answer in a few sentences.

4. Suppose instead that 0 < R < 1.

How would your answer to question 1(a) change?

How would your answer to question 1(b) change?

How would your answer to question 2(b) change?