 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/1/d/ii/solution

f (x) = c x

c^{2} x = (c + 1) x

for all

x

, so

c

c^{2} - c - 1

f (x) = \frac{1 + 5}{2} x or f (x) = \frac{1 - 5}{2} x .

For either choice,

c^{2} = c + 1

proves the required equality directly. The positive coefficient is the golden ratio.

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