Problem 572
Is it true that if \(\displaystyle f:\R\to\R\) is concave, then \(\displaystyle \lim\limits_{-\infty}f<\infty\) or \(\displaystyle \lim\limits_{\infty}f<\infty\)? Difficulty: 4. |
Problem 573
Is it true that if \(\displaystyle f:\R\to\R\) is convex and \(\displaystyle \lim\limits_{-\infty}f=-\infty\), then \(\displaystyle \lim\limits_{\infty}f=\infty\)? Difficulty: 4. |
Problem 575
Is it true that if \(\displaystyle f:\R\to\R\) is concave and \(\displaystyle \lim\limits_{-\infty}f\) is finite, then \(\displaystyle f\) is monotone decreasing? Difficulty: 4. |
Problem 576
Prove that if \(\displaystyle f:\R\to\R\) is additive, then \(\displaystyle f^2\) is weakly convex. Difficulty: 4. |
Problem 578
Prove that if \(\displaystyle f:\R\to\R\) is strictly monotone increasing and convex, then \(\displaystyle f^{-1}\) is concave on the interval \(\displaystyle (\inf f, \sup f)\). Difficulty: 4. |
Problem 571
Prove that if \(\displaystyle f:[a,b]\to\R\) is convex, then \(\displaystyle \lim\limits_{a+0}f\) and \(\displaystyle \lim\limits_{b-0}f\)exists and finite, moreover \(\displaystyle \lim\limits_{a+0}f\le f(a)\) and \(\displaystyle \lim\limits_{b-0}f\le f(b)\). Difficulty: 5. |
Problem 574
Prove that if \(\displaystyle f\) is weakly convex then \(\displaystyle f\left(\frac{x_1+\ldots+x_n}n\right)\le\frac{f(x_1)+\ldots+f(x_n)}n. \) Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |