MCQ
$f(x) = \left| {\left| x \right| - 1} \right|$ is not differentiable at
- A$0$
- B$ \pm 1,\,0$
- C$1$
- D$ \pm \,1$
✓
Answer
$ = \left\{ \begin{array}{l}|x| - 1,\,\,\,\,\,\,\,|x| - 1 \ge 0\\ - |x| + 1,\,\,\,|x| - 1 < 0\end{array} \right.$
$ = \left\{ \begin{array}{l}|x| - 1,\,\,\,x \le - 1\,\,{\rm{or}}\,x \ge 1\\ - |x| + 1,\,\,\,\,\,\,\, - 1 < x < 1\end{array} \right.$
$ = \left\{ \begin{array}{l} - x - 1,\,\,\,\,\,x \le - 1\\x + 1,\,\,\,\,\,\, - 1 < x < 0\\ - x + 1,\,\,\,\,\,0 \le x < 1\\\,\,x - 1,\,\,\,\,\,\,\,x \ge 1\end{array} \right.$
From the graph.
It is clear that $f(x)$ is not differentiable at $x = - 1,\,0$ and $1$.
$ = \left\{ \begin{array}{l}|x| - 1,\,\,\,x \le - 1\,\,{\rm{or}}\,x \ge 1\\ - |x| + 1,\,\,\,\,\,\,\, - 1 < x < 1\end{array} \right.$
$ = \left\{ \begin{array}{l} - x - 1,\,\,\,\,\,x \le - 1\\x + 1,\,\,\,\,\,\, - 1 < x < 0\\ - x + 1,\,\,\,\,\,0 \le x < 1\\\,\,x - 1,\,\,\,\,\,\,\,x \ge 1\end{array} \right.$
From the graph.
It is clear that $f(x)$ is not differentiable at $x = - 1,\,0$ and $1$.
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