A function f: R arrow R satisfies the equation f(x +y)=f(x) .f(y), x, y ∈ R and f(x) ≠ 0 for any x ∀ R. If f is differentiable at ' 0 ' f'(0)=3 and f(5)=2 then f'(5)=

Question

A function f: R arrow R satisfies the equation f(x +y)=f(x) .f(y), x, y ∈ R and f(x) ≠ 0 for any x ∀ R. If f is differentiable at ' 0 ' f'(0)=3 and f(5)=2 then f'(5)= (A) 2 (B) 4 (C) 6 (D) 8

Tardigrade Admin · Accepted Answer

f(x)=f(x+0)=f(x). f(0) ⇒ f(0)=1 f'(5)= displaystyle lim h arrow 0 (f(5+h)-f(5)/h) displaystyle lim h arrow 0 (f(5) f(h)-f(5)/h) = displaystyle lim h arrow 0 (f(5)[f(h)-1]/h) =f(5). displaystyle lim h arrow 0 (f(h)-f'(0)/h)=f(5) f(0) =2 × 3=6

Q. A function f:R→R satisfies the equation f(x+y)=f(x).f(y),x,y∈R and f(x)=0 for any x∀R. If f is differentiable at ' 0 ' f′(0)=3 and f(5)=2 then f′(5)=

2

4

6

8

f(x)=f(x+0)=f(x).f(0) ⇒f(0)=1 f′(5)=h→0lim​hf(5+h)−f(5)​ h→0lim​hf(5)f(h)−f(5)​ =h→0lim​hf(5)[f(h)−1]​ =f(5).h→0lim​hf(h)−f′(0)​=f(5)f(0) =2×3=6

Q. A function $f : R \to R$ satisfies the equation $f (x + y) = f (x) . f (y), x, y \in R$ and $f (x) \neq = 0$ for any $x \forall R$ . If $f$ is differentiable at ' 0 ' $f^{'} (0) = 3$ and $f (5) = 2$ then $f^{'} (5) =$