Sets, Relations & Functions
Functional Equation — Derivative Value
nta_pyq_2023_apr
Grade 11
Question:
For the differentiable function $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$, let $3f(x)+2f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10$, then $\left|f(3)+f'\!\left(\dfrac{1}{4}\right)\right|$ is equal to
\dfrac{33}{5}
8
\dfrac{29}{5}
13
Step-by-Step Solution
Key Concept: Replace $x\to\frac{1}{x}$: $3f(\frac{1}{x})+2f(x)=x-10$. Eliminate $f(\frac{1}{x})$: $5f(x)=\frac{3}{x}-2x-10$.
$f(3)=-3,\ f'(\frac{1}{4})=-10$. $|-3-10|=13$.
Correct Answer: 4