Left hand derivative and right hand derivative of a function `f(x)` at a point `x=a` are defined as
`f'(a^-)=lim_(h to 0^(+))(f(a)-f(a-h))/(h)`
`=lim_(hto0^(+))(f(a+h)-f(a))/(h)`
`andf'(a^(+))=lim_(h to 0^+)(f(a+h)-f(a))/(h)`
`=lim_(hto0^(+))(f(a)-f(a+h))/(h)`
`=lim_(hto0^(+))(f(a)-f(x))/(a-x)` respectively.
Let `f` be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function.
The statement `lim_(hto0)(f(-x)-f(-x-h))/(h)=lim_(hto0)(f(x)-f(x-h))/(-h)` implies that for all `x"inR`,

Options

A. `f` is odd
B. `f` is even
C. `f` is neither even nor odd
D. Nothing can be concluded

Answer

B