Prove that $f$ must have an inflection point at $0$

Since $f''$ is continuous and injective it is either increasing or decreasing.

Consider a sequence of points $x_k \searrow 0$. For each $k$ there exists $y_k \in (0,x_k)$ satisfying $$f''(y_k) = \frac{f'(x_k) - f'(0)}{x_k - 0} = \frac{f'(x_k)}{x_k} > 0.$$

Since $f''$ is continuous you can then show $f''(0) \ge 0$. Since $f''$ attains positive values along a sequence decreasing to $0$, it follows that $f''$ cannot be decreasing - it is thus increasing.

Can you proceed from there?