Applications of Derivatives
Derivative from first principles using functional equation
nta_pyq_2023_jan
Grade 12
Question:
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 $f'(0) = 2$, then $|f(-2)|$ is equal to ____.
Step-by-Step Solution
Key Concept: Use the functional equation to show $f'(x) = f'(0) = 2$ for all $x$, giving $f(x) = 2x + C$. Determine $C$ by setting $x = y = 0$: $f(0) = 2f(0) - 1 \Rightarrow f(0) = 1$.
Setting $x = y = 0$: $f(0) = 2f(0) - 1 \Rightarrow f(0) = 1$. Using the definition of derivative: $f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\to 0}\frac{f(h)-1}{h} = \lim_{h\to 0}\frac{f(h)-f(0)}{h} = f'(0) = 2$. So $f(x) = 2x + 1$. Thus $|f(-2)| = |-4+1| = |-3| = 3$.
Correct Answer: 3