Is there a $C^2$ function with these properties?

Question

$f(x)< 0$, $f'(x)<0$ and $f''(x)>0$ for all $x\in\mathbb{R}$

Answer 1

Since $f$ is convex ($ f''(x)>0 $,) we conclude that$$ \forall\, x, \qquad f(x)\ge f(0)+f'(0) x$$ But since $f'(0)<0$ this implies that $f(x)>0$ for $ x<-f(0)/f'(0)$ which is absurd.

So, the answer is no.