Let f: R arrow R be a differentiable function that satisfies the relation f(x+y)=f(x)+f(y)-1, ∀ x, y ∈ R. If f prime(0)=2, then |f(-2)| is equal to

Question

Let f: R arrow R be a differentiable function that satisfies the relation f(x+y)=f(x)+f(y)-1, ∀ x, y ∈ R. If f prime(0)=2, then |f(-2)| is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

f ( x + y )= f ( x )+ f ( y )-1 f prime(x)= displaystyle lim h arrow 0 (f(x+h)-f(x)/h) f prime(x)= displaystyle lim h arrow 0 (f(h)-f(0)/h)=f prime(0)=2 f prime(x)=2 ⇒ d y=2 d x y =2 x + C x =0, y =1, c =1 y =2 x +1 |f(-2)|=|-4+1|=|-3|=3

Q. 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 _____

f(x+y)=f(x)+f(y)−1 f′(x)=h→0lim​hf(x+h)−f(x)​ f′(x)=h→0lim​hf(h)−f(0)​=f′(0)=2 f′(x)=2⇒dy=2dx y=2x+C x=0,y=1,c=1 y=2x+1 ∣f(−2)∣=∣−4+1∣=∣−3∣=3

Q. Let $f : R \to R$ be a differentiable function that satisfies the relation $f (x + y) = f (x) + f (y) - 1, \forall x, y \in R$ . If $f^{'} (0) = 2$ , then $∣ f (- 2) ∣$ is equal to _____