Differentiation problem probably using increasing or decreasing property of differentiation

Question

Let $f:\mathbb{R} \to \mathbb{R}$ be twice differentiable function on $\mathbb{R}\setminus\{p\}$, for some $p$ belonging to $\mathbb{R}$. If $f'(x)<0<f''(x)$ on $x <p$ and $f'(x) >0 >f''(x)$ on $x>p$, then $f$ is not differentiable at $p$.

Answer 1

If $f$ is continuous at $p$ then it must have a strict minimum at $p$.

If $f$ is differentiable at $p$ then it must be continuous at $p$ and hence $f'(p) = 0$. Since $f''(x) < 0 $ for $x >p$ we must have $f'(x) <0$ for $x >p$, a contradiction.