MathDB

6

Part of 1998 IMC

Problems(2)

integral inequality

Source: IMC 1998 day 1 problem 6

11/1/2005
Let f:[0,1]→Rf: [0,1]\rightarrow\mathbb{R} be a continuous function satisfying xf(y)+yf(x)≤1xf(y)+yf(x)\le 1 for every x,y∈[0,1]x,y\in[0,1]. (a) Show that ∫01f(x)dx≤π4\int^1_0 f(x)dx \le \frac{\pi}4. (b) Find such a funtion for which equality occurs.
calculusintegrationinequalitiesfunctiontrigonometryreal analysisreal analysis unsolved
IMC 1998 Problem 12

Source: IMC 1998 Day 2 Problem 6

10/28/2020
f:(0,1)→[0,∞)f: (0,1) \rightarrow [0, \infty) is zero except at a countable set of points a1,a2,a3,...a_{1}, a_2, a_3, ... . Let bn=f(an)b_n = f(a_n). Show that if ∑bn\sum b_{n} converges, then ff is differentiable at at least one point. Show that for any sequence bnb_{n} of non-negative reals with ∑bn=∞\sum b_{n} =\infty , we can find a sequence ana_{n} such that the function ff defined as above is nowhere differentiable.
differentiabilityreal analysis