A professor has explained to my class St. Petersburgh paradox and has introduced the concept of utility function.
The professor then asked us to find an utility function with a positive first derivative and a negative second derivative to obtain a cost functional with a positive first derivative and a negative second derivative.
I have found this article. and the own Wikipedia page of the paradox, in which is said
For any unbounded utility function, one can find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger
It is clear that my professor is talking about a logarithm, whose first derivative is $\frac{1}{x}$, positive for all positive numbers, and the second is $-\frac{1}{x^2}$, negative for all real numbers. However, I have no idea on how to obtain an unbounded logarithm function.