Since $f''(x)>0$ for every $x$, the derivative $f'(x)$ is strictly increasing.
Therefore, $f'(x)=0$ can happen at most once, so A is true.
Also, if $f$ has a local minimum at $c$, then $f'(c)=0$.
Since $f'$ is strictly increasing, $f'(x)<0$ for $x<c$ and $f'(x)>0$ for $x>c$, so $f$ decreases before $c$ and increases after $c$.
Hence that local minimum is the unique global minimum, so B is true.
If $f'(1)<0<f'(4)$, then by continuity of $f'$, there is some $c\in(1,4)$ such that $f'(c)=0$. Since $f'$ is strictly increasing, this $c$ is unique, and it gives the unique global minimum. Thus C is true.
D is false because $f(x)=x+e^x$ satisfies $f''(x)=e^x>0$ and has no critical point, but $f(x)\to-\infty$ as $x\to-\infty$.
