Conjugate functions

1. Find $f^*(y)$, if $f(x) = ax + b$
2. Find $f^*(y)$, if $f(x) = -\log x, \;\; x\in \mathbb{R}_{++}$
3. Find $f^*(y)$, if $f(x) = e^x$
4. Find $f^*(y)$, if $f(x) = x \log x, x \neq 0, \;\;\; f(0) = 0, \;\;\; x \in \mathbb{R}_+$
5. Find $f^*(y)$, if $f(x) =\frac{1}{2} x^T A x, \;\;\; A \in \mathbb{S}^n_{++}$
6. Find $f^*(y)$, if $f(x) =\max\limits_{i} x_i, \;\;\; x \in \mathbb{R}^n$
7. Find $f^*(y)$, if $f(x) = -\dfrac{1}{x}, \;\; x\in \mathbb{R}_{++}$
8. Find $f^*(y)$, if $f(x) = -0,5 - \log x, \;\; x>0$
9. Find $f^*(y)​$, if $f(x) = \log \left( \sum\limits_{i=1}^n e^{x_i} \right)​$
10. Find $f^*(y)$, if $f(x) = - (a^2 - x^2)^{1/2}, \;\;\; \vert x\vert \le a, \;\;\; a>0$
11. Find $f^*(Y)$, if $f(X) = - \ln \det X, X \in \mathbb{S}^n_{++}$
12. Find $f^*(y)$, if $f(x) = \|x\|$
13. Find $f^*(y)$, if $f(x) = \dfrac{1}{2}\|x\|^2$
14. Name any 3 non-trivial facts about conjugate function.
15. Find conjugate function to the $f(x) = \dfrac{1}{x}, \;\; x \in \mathbb{R}_{++}$
16. Find conjugate function to the $f(x) = x^p, \;\; x \in \mathbb{R}_{++}, \;\; p>1$
17. Prove, that if $f(x_1, x_2) = g_1(x_1) + g_2(x_2)$, then $f^*(y_1, y_2) = g_1^*(y_1) + g_2^*(y_2)$
18. Prove, that if $f(x) = g(x-b)$, then $f^*(y) = b^\top y + g^*(y)$
19. Prove, that if $f(x) = \alpha g(x)$, then $f^*(y) = \alpha g^*(y/\alpha)$
20. Prove, that if $f(x) = g(Ax)$, then $f^*(y) = g^*(A^{-\top}y)$
21. Prove, that if $f(x) = \inf\limits_{u+v = x} (g(u) + h(v))$, then $f^*(y) = g^*(y) + h^*(y)$