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CGP EDU Academic Team
Published on: August 14, 2026
Let f(x m y n ) = mf(x) + nf(y) for all x, y ∈ R + and for all m, n ∈ R. If f ′ (x) exists and has the value 
, then find 
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
Sol. For any x ∈ R + , we have
f(x m y n ) = mf(x) + nf(y) ...(i)
∴ f(1) = f(1) + f(1)
[Putting x = y = m = n = 1]
⇒ f(1) = 0
f ′ (x) = 
= 
= 
= 
= 
∴ Putting y = 1 in (i), we get
f(x m ) = mf(x)
⇒ f {x 1/m ) m } = mf (x 1/m )
∴ f(x) = 
= 
= 
[ mf (x) = f(x m )]
= 
But, f ′ (x) = 
∴ 
= 
⇒ 
= e
⇒ 
= e, where θ = 
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