If `f(x)={{:(,x[x], 0 le x lt 2),(,(x-1)[x], 2 le x lt 3):}` where [.] denotes the greatest integer function, then continutity and diffrentiability of f(x)

Options

A. both f'(1) and f'(2) do not exist
B. f'(1) exist but f'(2) does not exist
C. f'(2) exist but f'(1) does not exist
D. both f'(1) and f'(2) exist

Answer

A

Explanation

We have
`f(x)={{:(,x xx 0=0,0 le x lt 1),(,x xx 1=x,1 le x lt2),(,(x-1)xx2=2(x-1),2le x lt 3):}`
Clearly, f(x) is not conintuous at x=1, So, it can not be differentiable at x=1.
We observe that f(x) is continuous at x=2, but it is not differentiable at x=2.
Hence, both f'(1) and f'(2) do not exist