Consider f(x) = [ 2 , , ( ,x , , - , , ^3 ,x ) , , + , , | ,x , ,… - Vidyadip

MCQ

Consider $f(x) =$ $\left[ {\frac{{2\,\,\left( {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right)\,\, + \,\,\left| {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right|}}{{2\,\,\left( {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right)\,\, - \,\,\left| {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right|}}} \right]$ , $x \ne \,\frac{\pi}{2} \,for\, x \in (0, \pi ) f(\pi /2) = 3$

where [ ] denotes the greatest integer function then,

✓
$f$ is continuous $\&$ differentiable at $x = \pi /2$
B
$f$ is continuous but not differentiable at $x = \pi /2$
C
$f$ is neither continuous nor differentiable at $x = \pi /2$
D
none of these

✓

Answer

Correct option: A.

$f$ is continuous $\&$ differentiable at $x = \pi /2$

a
In the immediate neighborhood of

$x = \pi /2 , sinx > sin^3x$

$ \Rightarrow |sinx - sin^3x| = sinx - sin^3x$
Hence for $x\, \ne \,\pi /2 $ ,

$f (x) =$ $\left[ {\frac{{2(\sin x - {{\sin }^3}x)\, + \sin x - {{\sin }^3}x}}{{2(\sin x - {{\sin }^3}x)\, - \sin x + {{\sin }^3}x}}} \right]$ =$\frac{{3\sin x\, - \,3{{\sin }^3}x}}{{\sin x - {{\sin }^3}x}}\, = \,3$

Hence $f$ is continuous and diff. at $x = \pi /2$

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