 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/4/b/solution

The second-order remainder form of Taylor's theorem gives, for a point

ξ_{x}

between

0

and

x

f (x) = - 5 + 3 x + \frac{1}{2} f^{''} (ξ_{x}) x^{2} .

For

x \neq = 0

, the given curvature bounds imply

- 5 + 3 x + x^{2} \leq f (x) < - 5 + 3 x + \frac{3}{2} x^{2} .

In particular

f (1) < - 1/2 < 0

and

f (2) \geq 5 > 0

(1, 2)

f^{'}

gives

f^{'} (x) = 3 + x f^{''} (η_{x}) \geq 3 + 2 x > 0

for

x \in (1, 2)

. Thus

f

is strictly increasing there, and it cannot have two roots. There is exactly one root in $(1, 2)$ .

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