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 $p$.
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The implicit differentiation tag seems misplaced? – copper.hat Sep 01 '17 at 04:06
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Sir, I was aware that the tag is wrong but when I was typing differentiation it is the only tag which appeared and my apologies for tagging it wrong – user476538 Sep 02 '17 at 11:53
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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.
copper.hat
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