karooooooooooooo 4

first the options look horrible and u can't check by putting values,,,,,,,so put t=2x

now it looks as
f(x)=sin⁴ t/2 + cos⁴ t/2
ofcourse the options simplify too...

now as usual diffrentiateand ultimately u should get

f^|(t)=sin t (2sin²t/2 - 1)

Now all we hav to find is where f^|(t) is positive and then put t=2x.....i think the prob is now solved

f^|(x)=4sin³x cosx -4cos³x sinx

=4sinxcosx(sin²x=cos²x)
=-2sin2x(cos2x)
=-sin4x

-sin4x>0 (as f^|(x) is increasing)

sin4x<0

pi<4x<2(pi)

pi/4<x<pi/2

only option (b) lies in the range therefore (b) is the ans