### Differentiation Question 3

#### Question 3 - 29 January - Shift 1

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

## Show Answer

#### Answer: 3

#### Solution:

#### Formula: Differentiability at a point, Modulus of a function

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

$f^{\prime}(x)=\lim _{h \to 0} \frac{f(x+h)-f(x)}{h}$

$f^{\prime}(x)=\lim _{h \to 0} \frac{f(h)-f(0)}{h}=f^{\prime}(0)=2$

$f^{\prime}(x)=2 \Rightarrow d y=2 d x$

$y=2 x+C$

$x=0, y=1, c=1$

$y=2 x+1$

$|f(-2)|=|-4+1|=|-3|=3$