Zoom Recitation 15. 2 | THU-QZC-Creative Problem Solving, Fall 2022
Title: Real analysis V (part 3)
Lecturer: Cezar Lupu, Yanqi Lake BIMSA & YMSC, Tsinghua University
Description: In this recitation we discuss about the anatomy of a Putnam problem. In this talk, I shall bring into perspective an analysis problem given at the Putnam Mathematical Competition in 2007. The question B2 on the exam reads as the following:
Suppose that f:[0,1]→R has a continuous derivative and that ∫_0^1 f(x)dx=0.
Prove that for every α∈(0,1),
∣∣∣∫_0^{\alpha}f(x)dx∣∣∣≤1/8 sup0≤x≤1|f′(x)|.
Following Polya's approach on how to solve a problem (https://math.berkeley.edu/~gmelvin/po..., we "dissect" the above question by giving several solutions and by relating it to other similar problems or problems with similar ideas. In fact, we go back to one of Polya's original problems and show how ideas can be applied in our situation. Last but not least, we also discuss some variations of our main question.