# If $f''(x)$ is increasing, unbounded and has a unique root. What can we say about the function?

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?

• $f(x)$ should have a unique root – Vasya Mar 20 '18 at 17:26
• As you noticed, $f''(x)$ positive means $f'(x)$ is increasing from the inflection point on. It also means that the slope of the tangent line to the function is increasing. – Vasya Mar 20 '18 at 18:02
• @Vasya and that means that $f(x)$ is unbounded and also has a unique root? – D F Mar 20 '18 at 19:11