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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
MCQ
If $f(x) = \left\{ \begin{array}{l}x,\,\,\,\,\,\,\,\,\,\,\,0 \le x \le 1\\2x - 1,\,\,\,1 < x\end{array} \right.$, then
  • A
    $f$ is discontinuous at$x = 1$
  • B
    $f$ is differentiable at $x = 1$
  • C
    $f$ is continuous but not differentiable at $x = 1$
  • D
    None of these
✓

Answer

$f(x) = \left\{ \begin{array}{l}\,\,\,x,\,\,\,\,\,\,\,\,0 \le x \le 1\\2x - 1,\,\,\,\,\,\,x > 1\end{array} \right.$
$\mathop {\lim }\limits_{x \to {1^ - }} f(x) = \mathop {\lim }\limits_{h \to 0} f(1 - h) = \mathop {\lim }\limits_{h \to 0} (1 - h) = 1$
$\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{h \to 0} f(1 + h) = \mathop {\lim 2}\limits_{h \to 0} (1 + h) - 1 = 1$
$\because \,\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=1$
$\therefore$ Function is continuous at $x = 1$.
$Lf'(1) = \mathop {\lim }\limits_{h \to 0} \frac{{f(1 - h) - f(1)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \frac{{(1 - h) - 1}}{{ - h}} = 1$
$Rf'(1) = \mathop {\lim }\limits_{h \to 0} \frac{{f(1 + h) - f(1)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \frac{{2 + 2h - 1 - 1}}{h} = 2$
$\therefore Lf'(1) \ne Rf'(1)$
$\therefore$  Function is not differentiation at $x = 1$

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