The function, `f(x) = cos^(-1) (cos x)` is

Options

A. discontinuous at infinitely many-points
B. everywhere differentiable such that f'(x)=1
C. not differentiable at `x=n pi, n in Z and f'(x)=1, x ne n pi`
D. not differentiable at `x=n pi, n in Z` and `f'(x)=(-1)^(n), x in (n pi,(n+1)pi), n in Z`

Answer

D

Explanation

The graph of the `f(x)=cos^(-1)` (cos x) as given in Fig.6
It is evident from the curve y=f(x) that f(x) is everywhere continuous but it is not differentiable `x=n, n in Z`
Also, `f'(x)=(-1)^(n), x in (n pi,(n+1)pi),"where "n in Z`
We can observe that f(x) is an even periodic function with period `2pi`.