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Question

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as
f(a)=limh0+f(a)f(ah)h=limh0f(a)f(ah)h=limxa+f(a)f(x)ax respectively
Let f be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

The statement limh0f(x)f(xh)h=limh0f(x)f(xh)h implies that for all x ϵR

A
f is odd
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B
f is even
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C
f is neither odd nor even
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D
nothing can be concluded
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Solution

The correct option is B f is even
limh0f(x)f(xh)h=limh0f(xh)f(x)h
=f(x) ......(i)
and limh0f(x)f(xh)h=f(x) ......(ii)
from (i) and (ii) f(x) is odd function and hence f(x) is even function.

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