Let f:R R be a function defined by f(x) = ,(x, , x^3 ). The set o… - Vidyadip

MCQ

Let $f:R \to R$ be a function defined by $f(x) = \max \,(x,\,{x^3}).$ The set of all points where $f(x)$ is not differentiable is

A
$\{ - 1,\,1\} $
B
$\{ - 1,\,\,0\} $
C
$\{ 0,\,\,1\} $
✓
$\{ - 1,\,\,0,\,1\} $

✓

Answer

Correct option: D.

$\{ - 1,\,\,0,\,1\} $

d
(d) If $x < - 1$ then $x > {x^3}$. So, $f(x) = x$,
If $x = - 1$ then $x = {x^3}$. So, $f(x) = x$,
If $ - 1 < x < 0$ then $x < {x^3}$. So, $f(x) = {x^3}$,
If $x = 0$ then $x = {x^3}$. So, $f(x) = {x^3}$,
If $0 < x < 1$ then $x > {x^3}$. So, $f(x) = x$,
If $x = 1$ then $x = {x^3}$. So, $f(x) = x$,
If $x > 1,$ then $x < {x^3}$. So, $f(x) = {x^3}$
Thus $f(x) = x,\,x \le - 1$,$f(x) = \left\{ {\begin{array}{*{20}{c}}x&,&{x \le - 1}\\{{x^3}}&,&{ - 1 < x \le 0}\\x&,&{0 < x \le 1}\\{{x^3}}&,&{x > 1}\end{array}} \right.$
Clearly, $f(x)$ is not differentiable at $x = - 1,\,0,\,1$.

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