Consider f(x)= 2sin⁡x−sin3⁡x+sin⁡x−sin3⁡x2sin⁡x−sin3⁡x−sin⁡x−sin3⁡x ,x≠π/2 for x∈0,π and f(π/2)=3Where [ ] denotes the greatest integer function, then

Given f(x)=2sin⁡x(1−sin⁡x)(1+sin⁡x)+|sin⁡x||1−sin⁡x||1+sin⁡x|2sin⁡x(1−sin⁡x)(1+sin⁡x)−|sin⁡x||1−sin⁡x||1+sin⁡x|=31=3 for x∈(0,π);x≠π/2Also f(π/2)=3 (given)  f is continuous at x=π/2 and also differentiable at x=π/2 (as f is a constant function).

Consider f(x)= 2sin⁡x−sin3⁡x+sin⁡x−sin3⁡x2sin⁡x−sin3⁡x−sin⁡x−sin3⁡x ,x≠π/2 for x∈0,π and f(π/2)=3Where [ ] denotes the greatest integer function, then

Given f(x)=2sin⁡x(1−sin⁡x)(1+sin⁡x)+|sin⁡x||1−sin⁡x||1+sin⁡x|2sin⁡x(1−sin⁡x)(1+sin⁡x)−|sin⁡x||1−sin⁡x||1+sin⁡x|=31=3 for x∈(0,π);x≠π/2Also f(π/2)=3 (given)  f is continuous at x=π/2 and also differentiable at x=π/2 (as f is a constant function).

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Consider $f (x) =$ $[\frac{2 (\sin x - \sin^{3} x) + |\sin x - \sin^{3} x|}{2 (\sin x - \sin^{3} x) - |\sin x - \sin^{3} x|}]$
$, x \neq π / 2 for x \in (0, π) and f (π / 2) = 3$
Where $[]$ denotes the greatest integer function, then

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f is discontinuous at $x = π / 2$

f is continuous but not differentiable at $x = π / 2$

f is continuous and differentiable at $x = π / 2$

f is neither continuous nor differentiable at $x = π / 2$

answer is A.

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Detailed Solution

Given $f (x) = [\frac{2 \sin x (1 - \sin x) (1 + \sin x) + | \sin x | | 1 - \sin x | | 1 + \sin x |}{2 \sin x (1 - \sin x) (1 + \sin x) - | \sin x | | 1 - \sin x | | 1 + \sin x |}]$
$= [\frac{3}{1}] = 3 for x \in (0, π); x \neq π / 2$
Also $f (π / 2) = 3$ (given)  f is continuous at $x = π / 2$ and also differentiable at $x = π / 2$ (as f is a constant function).