 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/6/a/solution

Repeated use of the product rule and chain rule gives

f^{'} (x) = \frac{- 2 x}{( 1 + x ^{2} ) ^{2}}, f^{''} (x) = \frac{2 ( 3 x ^{2} - 1 )}{( 1 + x ^{2} ) ^{3}}, f^{'''} (x) = \frac{24 x ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{4}} .

The denominators are strictly positive on the whole real line, making the signs in the next parts easy to read.

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