Continuity and DifferentiabilityHard
Question
If f(x - y), f(x).f(y) and f(x + y) are in A.P. ∀ x, y ∈ R and f(0) ≠ 0, if f(x) has continuous derivative for x ∈ R, then -
Options
A.
f(x)dx 0
f(x)dx 0B.f(x)dx = 2 
f(x)dx
f(x)dx = 2 f(x)dxC.f′(x)dx = 0
f′(x)dx = 0D.f′(3) + f′(-3) = 0
Solution
f(x - y), f(x)f(y), f(x + y) are in A.P.
⇒ 2f(x)f(y) = f(x - y) + f(x + y) ......(i)
Now put x = 0 in (i)
f(y) + f(-y) = 2f(y) ⇒ f(y) = f(-y)
⇒ f(x) is an even function
⇒ f′(x) is an odd function
⇒ f′(-3) = - f′(3) ⇒ f′(3) + f′(-3) = 0
⇒ 2f(x)f(y) = f(x - y) + f(x + y) ......(i)
Now put x = 0 in (i)
f(y) + f(-y) = 2f(y) ⇒ f(y) = f(-y)
⇒ f(x) is an even function
⇒ f′(x) is an odd function
⇒ f′(-3) = - f′(3) ⇒ f′(3) + f′(-3) = 0
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