MATH100 December 2018
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q6 • Q7 • Q8 • Q9 • Q10 • Q11 (a) • Q11 (b) •
Question 11 (b)
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 .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
If we can find and such that , then according to Rolle's theorem, there exists such that .
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
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 , 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 .
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag