Differentiable function from a functional equation in f(x + y)
This type of question, asked 4 times: 2021 Paper 10, 26 Jun 2022, 29 Jan 2023, 22 Jan 2025. All 4 questions below, oldest first.
Try each one first, then open its answer.
Question 1
2021 Paper 10
87
Let f:R→R satisfy the equation f(x+y)=f(x)⋅f(y) for all x,y∈R and f(x)=0 for any x∈R. If the function f is differentiable at x=0 and f′(0)=3, then limh→0h1(f(h)−1) is equal to
Question 2
26 Jun 2022
171
Let f:R→R satisfy f(x+y)=2xf(y)+4yf(x),∀x, y∈R. If f(2)=3, then 14⋅f′(2)f′(4) is equal to
Question 3
29 Jan 2023
82
Let f:R→R be a differentiable function that satisfies the relation f(x+y)=f(x)+f(y)−1,∀x,y∈R If f′(0)=2, then ∣f(−2)∣ is equal to
Question 4
22 Jan 2025
66
Let f(x) be a real differentiable function such that f(0)=1 and f(x+y)=f(x)f′(y)+f′(x)f(y) for all x,y∈R. Then ∑n=1100logef(n) is equal to :
