A RESEARCHER IS MODELLING THE EVOLUTION OF A COMPANY'S MARKET SHARE IN TIME USING THE FUNCTION F(X] = RX! 1 + MK. 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. (E) 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 2' 1} 3. (A) IT MAY BE SHOWN THAT F ' (X) = R (1+MX) 2, WHICH IS ALWAYS POSITIVE. DESCRIBE IN A PARAGRAPH WHAT THE FEATURE F ' (X) 2' 0 FOR ALL X E D IMPLIES, IN TERMS OF MARKET SHARES; (B) WHY MIGHT THE FEATURE F ' (X) 2' 0 FOR ALL )1 2 0 BE UNSUITABLE FOR MODELLING CERTAIN MARKET SITUATIONS'? DESCRIBE YOUR ANSWER IN A FEW SENTENCES. 4. SUPPOSE INSTEAD THAT 0