If f(x) = * 20 c x [ x ], , , , , , , , , , , , , , & 0 x < 2 ( x… - Vidyadip

MCQ

If $f(x) = \left\{ {\begin{array}{*{20}{c}}
{x\left[ x \right],\,\,\,\,\,\,\,\,\,\,\,\,\,}&{0 \le x < 2}\\
{\left( {x - 1} \right)\left[ x \right]\,,\,\,\,}&{2 \le x \le 4}
\end{array}} \right.,$ where $[.]$ denotes greatest integer function, then

✓
neither $f'(1)$ exist nor $f'(2)$ exist
B
$f'(1)$ exists but $f'(2)$ does not exist
C
$f'(2)$ exists but $f'(1)$ does not exist
D
both $f'(1)$ as well as $f'(2)$ exist.

✓

Answer

Correct option: A.

neither $f'(1)$ exist nor $f'(2)$ exist

a
$f(x)=\left\{\begin{array}{ll}{0,} & {x \in[0,1)} \\ {x,} & {x \in(1,2]} \\ {2(x-1),} & {x \in(2,3)}\end{array}\right.$

$f(\mathrm{x})$ is discontinuous at $\mathrm{x}=1,$ hence it is not differentiable at that point.

At $x=2$

$\mathrm{LHD}=1$ and $\mathrm{RHD}=2$

$\Rightarrow f(\mathrm{x})$ is not differentiable at $\mathrm{x}=2$

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