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
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Answer: 3
Solution:
Formula: Differentiability at a point, Modulus of a function
$f(x+y)=f(x)+f(y)-1$
$f^{\prime}(x)=\lim _{h \to 0} \frac{f(x+h)-f(x)}{h}$
$f^{\prime}(x)=\lim _{h \to 0} \frac{f(h)-f(0)}{h}=f^{\prime}(0)=2$
$f^{\prime}(x)=2 \Rightarrow d y=2 d x$
$y=2 x+C$
$x=0, y=1, c=1$
$y=2 x+1$
$|f(-2)|=|-4+1|=|-3|=3$
