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Left hand derivative and right hand deri...

Left hand derivative and right hand derivative of a function `f(x)` at a point `x=a` are defined as
`f'(a^-)=lim_(hrarr0^(+))(f(a)-f(a-h))/(h)`
`=lim_(hrarr0^(+))(f(a+h)-f(a))/(h)`
`andf'(a^(+))=lim_(hrarr0^(+))(f(a+h)-f(a))/(h)`
`=lim_(hrarr0^(+))(f(a)-f(a+h))/(h)`
`=lim_(hrarr0^(+)) (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.
If `f` is even function, which of the following is right hand derivative of `f'` at `x=a?`

A

(a) `underset(hrarr0^(-))lim(f'(a)+f'(-a+h))/(-h)`

B

(b) `underset(hrarr0^(-))lim(f'(a)+f'(-a-h))/(h)`

C

(c) `underset(hrarr0^(-))lim(-f'(-a)+f'(-a-h))/(-h)`

D

(d) `underset(hrarr0^(+))lim(f'(a)+f'(-a+h))/(-h)`

Text Solution

Verified by Experts

The correct Answer is:
A
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