Let f : be a differentiable function that satisfies the relation f ( x + y ) = f ( x ) + f ( y ) - 1 ,x , y . If f ' ( 0 ) = 2 , then | f ( - 2 ) | is equal to _______ . [2023]

Question

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 _______ . [2023]

Smart Achievers · Accepted Answer

(3)

$Given f (x + y) = f (x) + f (y) - 1$

$Putting x = y = 0 \Rightarrow f (0) = 1$

$f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$f' (0) = \lim_{h \to 0} \frac{f (h) - f (0)}{h} \Rightarrow f' (0) = 2$

$f' (x) = 2 \Rightarrow f (x) = 2 x + C \Rightarrow y = 2 x + C$

$Now, f (0) = 1$

$∴$ $1 = 2 (0) + C \Rightarrow C = 1$

So, $f (x) = 2 x + 1$

$| f (- 2) |$ $=$ $| 2 (- 2) + 1 |$ $=$ $| - 3 |$ $= 3$

Solution

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 _______ . [2023]

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Solution

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 _______ . [2023]

Ans. (3) Given f(x+y)=f(x)+f(y)-1 Putting x=y=0⇒f(0)=1 f'(x)=limh→0f(x+h)-f(x)h f'(0)=limh→0f(h)-f(0)h⇒f'(0)=2 f'(x)=2⇒f(x)=2x+C⇒y=2x+C Now, f(0)=1 ∴ 1=2(0)+C⇒C=1 So, f(x)=2x+1 |f(-2)|=|2(-2)+1|=|-3|=3

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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 _______ . [2023]