Science:Math Exam Resources/Courses/MATH100/December 2018/Question 11 (b)
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Question 11 (b) |
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Let be a twice differentialble function with a continuous second derivative and also, satisfying the property that
for each positive real number . We let . Prove that there exist infinitely many real numbers such that . |
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Hint |
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If we can find and such that , then according to Rolle's theorem, there exists such that . |
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Solution |
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Please rate my easiness! It's quick and helps everyone guide their studies. As in problem 11(a), we again set , then , so we only need to prove there are infinitely many such that . From 11(a), there are infinitely many such that , so we can choose three of them denoted as , such that . Then according to Rolle's theorem, there exists , , such that , . We use Rolle's theorem again, then there exists such that . There are infinitely triple groups of and with no overlapping region of , so there are infintely many such that . |