If f(x) = l , , , , , , , , , , , ,1, , ,x < 0 1 + x, , ,0 x < 2 … - Vidyadip

MCQ

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,1,\,\,x < 0\\1 + \sin x,\,\,0 \le x < \frac{\pi }{2}\end{array} \right.$ then $f'(0) = $

A
$1$
B
$0$
C
$\infty $
✓
Does not exist

✓

Answer

Correct option: D.

Does not exist

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

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

Hence, $f'(0)$ does not exist.

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