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

MCQ

Let $f(x) = \left\{ \begin{array}{l}1\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\forall x < 0\\1 + \sin x,\,\,\,\forall 0 \le x \le \pi /2\end{array} \right.$, then what is the value of $f'(x)$ at $x = 0$

A
$1$
B
$-1$
C
$\infty $
D
does not exist

✓

Answer

$f(x) = \left\{ \begin{array}{l}1\forall x < 0\\1 + \sin x ,\forall 0 \le x < \frac{\pi }{2}\end{array} \right.$
$\therefore f'(x) = \left\{ \begin{array}{l}0,\forall x < 0({\rm{LHD}})\\\cos x,0 \le x \le \pi /2,({\rm{RHD}})\end{array} \right.$
$\therefore f'(0) = \left\{ \begin{array}{l}0,x < 0\\\cos 0 = 1\end{array} \right.$,
$\therefore f'(0)$ does not exist.

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