Question

For x€ R f(x). = |log2 -sinx| and g(x) = f(f(x)) then

1) g is not differentiable at x= 2
2)g'(0) = cos(log2)
3)g'(0) = -cos(log2)
4) g is differentiable at x= 0 and g'(0) = -sin(log2)

Answer 1

Hello Dear
we have f(x) = |log2 - sinx|. { given
for x→0 log2 > sin x
since log 2 = 0.301
Therefore f(x) = log2 - sinx. { |x| = x if x>0
g(x) = log2 - sin(f(x))
=> log2 - sin(log2 -sinx)
Now from here we can conclude that g(x) is differentiable at x = 0 since sinx is diff. at x

so g'(x) =. 0 -cos(log2 -sinx)(-cosx)
=. cosx(cos(log2 - sinx))
so
g'(0) = cos(log2)
hence
option 2 is correct.