Problem 805
Suppose that \(\displaystyle f:\R\to\R\) is convex, \(\displaystyle f(5)=12\) and \(\displaystyle \alpha=\lim_{x\to\infty}f(x)\). What are the possible values of \(\displaystyle \alpha\)? Difficulty: 3. |
Problem 807
Find the maximal intervals for which the following functions are convex or concave. (1) \(\displaystyle e^x\), (2) \(\displaystyle \log x\), (3) \(\displaystyle |x|\), (4) \(\displaystyle x^a\) (\(\displaystyle a\in\R\)), (5) \(\displaystyle a^x\) (\(\displaystyle a>0\)) (6) \(\displaystyle \sin x\). Difficulty: 4. |
Problem 809
Is it true that the inverse of a convex function is concave? Difficulty: 4. |
Problem 808
\(\displaystyle f:(a,b)\to\R\) is convex, \(\displaystyle \psi:f(a,b)\to\R\) is convex and monotone increasing. Prove that in this case \(\displaystyle \psi\circ f\) is also convex. Difficulty: 5. |
Problem 806
In how many points the graphs of two convex functions can intersect? And a convex and a concave? Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |