Suppose $f(x)$ is a twice differentiable on $\mathbb{R}$ function. And $f(0), f'(0), f''(0)$ are negative. Also, the second derivative $f''(x)$ is known to satisfy the following conditions:
(i) it increases on $[0, +\infty)$
(ii) it has a unique root on $[0, +\infty)$
(iii) it's unbounded on $[0, +\infty)$
Which of these conditions does $f(x)$ satisfy?
Assume $f''(x) < 0 $ for $x \in [0, a), f''(a) = 0, f''(x) > 0$ for $x \in (a, +\infty)$. All that I've managed to infer is that on $[0, a) \ f'(x)$ decreases, at $x = a$ it has a local minimum and then starts to increase, hence $f(x)$ decreases firstly and has an inflecion point at $x = a$.
What else can we say about the function $f(x)$? Could you please give me any hints?
Thanks in advance!