Science:Math Exam Resources/Courses/MATH100/December 2018/Question 11 (b)

MATH100 December 2018

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Question 11 (b)
Let $g(x)$ be a twice differentialble function with a continuous second derivative and also, satisfying the property that ${\frac {x}{2}}\leq g(x)\leq {\frac {x}{2}}+1$ for each positive real number $x$ . We let $f(x)=g(x)+\sin(x)$ . Prove that there exist infinitely many real numbers $c$ such that $f''(c)=0$ .

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Hint
If we can find $x_{1}$ and $x_{2}$ such that $f'(x_{1})=f'(x_{2})$ , then according to Rolle's theorem, there exists $c\in (x_{1},x_{2})$ such that $f''(c)=0$ .

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. As in problem 11(a), we again set $h(x)=f(x)-{\frac {x+1}{2}}$ , then $f''(x)=h''(x)$ , so we only need to prove there are infinitely many $c$ such that $h''(c)=0$ . From 11(a), there are infinitely many $c$ such that $h(c)=0$ , so we can choose three of them denoted as $c_{1}<c_{2}<c_{3}$ , such that $h(c_{1})=h(c_{2})=h(c_{3})=0$ . Then according to Rolle's theorem, there exists $d_{1}\in (c_{1},c_{2})$ , $d_{2}\in (c_{2},c_{3})$ , such that $h'(d_{1})=0$ , $h'(d_{2})=0$ . We use Rolle's theorem again, then there exists $e\in (d_{1},d_{2})$ such that $h''(e)=0$ . There are infinitely triple groups of $c_{1},c_{2}$ and $c_{3}$ with no overlapping region of $(c_{1},c_{3})$ , so there are infintely many $e\in (c_{1},c_{3})$ such that $h''(e)=0$ .