Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 5 / a / i

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- Solution i

Solution

The first three derivatives factor as

f^{'} (x) = 2 x^{3} (3 x^{2} - 10), f^{''} (x) = 30 x^{2} (x^{2} - 2), f^{'''} (x) = 120 x (x^{2} - 1) .

Let

a = 10/3

. Sign analysis of the factors gives

f^{'} (x) > 0 f^{''} (x) > 0 f^{'''} (x) > 0 ⟺ x \in (- a, 0) \cup (a, \infty), ⟺ x \in (- \infty, - 2) \cup (2, \infty), ⟺ x \in (- 1, 0) \cup (1, \infty) .

All displayed endpoints are excluded because a factor vanishes there; the extra zero of

f^{''}

at

0

does not create a positive interval.

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