Is there a function $f$ such that $f(x)<0, f'(x)<0,f''(x)>0 \,\forall x\in \mathbb R$ ?
I think there is, since it would be quite strange that there is not (and maybe hard to prove). After trying some logarithms, exponentials and polynomials I am out of ideas.
Definitely polynomials doesn't work: If $f(x)$ is a polynomial such that $f(x)<0 \forall x\mathbb \in \mathbb R$ then it is clear that the leading exponent must be even and the coefficient of it has to be negative. But then after derivating we violate our initial condition for $f'(x)$. Any ideas?
Note: Given the level level of the other questions in the book, the most likely functions that a simple example would use are logarithms and exponentials, trigonometric functions and their inverses, and polynomials. However, any example is welcome.