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 $
✓
does not exist

✓

Answer

Correct option: D.

does not exist

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

$\therefore \,\,f'(x) = \left\{ \begin{array}{l}\,\,\,0,\,\,\,\,\forall \,x < 0\,({\rm{LHD}})\\\cos x,\,\,0 \le x \le \pi /2,\,\,({\rm{RHD}})\end{array} \right.$

$\therefore \,f'(0) = \left\{ \begin{array}{l}\,\,0\,\,\,\,,\,\,x < 0\\\cos 0 = 1\end{array} \right.$,

$\therefore \,f'(0)$ does not exist.

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