Problem 397
Find the inverse of \(\displaystyle f(x)=\frac{2x-3}{3x-2}\) on \(\displaystyle \R\setminus\{\tfrac23\}\). Difficulty: 1. |
Problem 407
Does there exist a function \(\displaystyle f:(0,1)\to \R\) which is bounded, but has no maximum? Difficulty: 1. |
Problem 386
Show that the following functions are injective on the given set \(\displaystyle H\), and calculate the inverse. (1) \(\displaystyle f(x)=3x-7, \ H=\R\); (2) \(\displaystyle f(x)=x^{2}+3x-6, \ H=[-3/2,\infty)\). Difficulty: 2. |
Problem 387
Show that the following functions are injective on the given set \(\displaystyle H\), and calculate the inverse. (1) \(\displaystyle f(x)=\frac{x}{x+1}, \ H=[-1,1]\); (2) \(\displaystyle f(x)=\frac{x}{|x|+1}, \ H=\R.\) Difficulty: 2. Solution is available for this problem. |
Problem 398
Are the following functions injective on \(\displaystyle [-1,1]\)? a) \(\displaystyle \displaystyle f(x)=\frac{x}{x^2+1}\), b) \(\displaystyle \displaystyle g(x)=\frac{x^{2}}{x^2+1}\). Difficulty: 2. Solution is available for this problem. |
Problem 401
Prove that all function \(\displaystyle f:\R \to \R\) can be obtained as the sum of an even and an odd function. Difficulty: 2. |
Problem 403
Let \(\displaystyle f(x)=x^3\) if \(\displaystyle x\) is rational, and \(\displaystyle f(x)=-x^3\) if \(\displaystyle x\) is irrational. Does \(\displaystyle f(x)\) have a unique inverse on \(\displaystyle (-\infty,+\infty )\)? Difficulty: 2. |
Problem 406
Prove that if \(\displaystyle f\) is strictly convex on the interval \(\displaystyle I\), then every line intersects the graph of \(\displaystyle f\) in at most 2 points. Difficulty: 2. |
Problem 408
Does there exist a function \(\displaystyle f:[0,1]\to \R\) which is bounded, but has no maximum? Difficulty: 2. |
Problem 412
Prove that if \(\displaystyle a_{1},...,a_{n}\geq 0\) and \(\displaystyle k>1\) is an integer, then \(\displaystyle \frac{a_{1}+...+a_{n}}{n}\leq \sqrt[k]{\frac{a_{1}^{k}+...+a_{n}^{k}}{n}}\). Difficulty: 3. |
Problem 389
Construct a non-constant periodic function with arbitrarily small periods. Difficulty: 4. |
Problem 405
Let \(\displaystyle f(x)=\max\{x,1-x,2x-3\}\). Is it monotone, or convex? Difficulty: 4. |
Problem 409
Does there exist a monotone function \(\displaystyle f\) such that (1) \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=(0,1);\phantom{\,\cup [2,3]}\) (2) \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1]\cup [2,3]\); Difficulty: 4. Solution is available for this problem. |
Problem 414
Prove that if \(\displaystyle g:A\to B\) and \(\displaystyle f:B\to C\) are convex, and \(\displaystyle f\) is monotone increasing, then \(\displaystyle f\circ g\) is convex. Difficulty: 4. |
Problem 415
Prove that if \(\displaystyle f\) is convex, then it can be obtained as the sum of a monotone increasing and a monotone decreasing function. Difficulty: 4. |
Problem 411
Prove that \(\displaystyle x^{k}\) is strictly convex on \(\displaystyle [0,\infty)\), for all \(\displaystyle k>1\) integer. Difficulty: 5. |
Problem 388
Find a function \(\displaystyle f:[-1,1]\to[-1,1]\) such that \(\displaystyle f(f(x))=-x \forall x\in[-1,1]\). Difficulty: 7. Solution is available for this problem. |
Problem 417
Can we obtain the function \(\displaystyle x^2\) as a sum of two periodic functions? Difficulty: 7. |
Problem 410
Does there exist a function which attains every real values on any interval? Difficulty: 8. |
Problem 418
Can we obtain the function \(\displaystyle x^2\) as a sum of three periodic functions? Difficulty: 10. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |