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STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,{e^x};\,\,\,\,x \le 0\\|1 - x|;\,\,x > 0\end{array} \right.$, then
  • A
    $f(x)$ is continuous at $x = 1$
  • B
    $f(x)$ is continuous at $x = 0$
  • ✓
    $(a)$ and $(b)$ both
  • D
    None of these
✓

Answer

Correct option: C.
$(a)$ and $(b)$ both
c
(c) $f(x) = \left\{ \begin{array}{l}{e^x}\,\,;\,\,\,x \le 0\\1 - x;\,\,0 < x \le 1\\x - 1\,;\,\,x > 1\end{array} \right.$

$Rf'(0) = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 + h) - f(0)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{1 - h - 1}}{h} = - 1$

$Lf'(0) = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 - h) - f(0)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \frac{{{e^{ - h}} - 1}}{{ - h}} = 1$

So, it is not differentiable at $x = 0$.

Similarly, it is not differentiable at $x = 1$.

But it is continous at $x = 0$, $1$.

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