Question:
If $x \neq 0$ and $f(x)$ satisfies $8f(x) + 6f\left(\dfrac{1}{x}\right) = x + 5$, then $\dfrac{d}{dx} \left(x^2 f(x)\right)$ at $x = 1$ is
If $x \neq 0$ and $f(x)$ satisfies $8f(x) + 6f\left(\dfrac{1}{x}\right) = x + 5$, then $\dfrac{d}{dx} \left(x^2 f(x)\right)$ at $x = 1$ is
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Use implicit differentiation and substitution to solve functional equations involving derivatives.
Updated On: May 19, 2025
- $-\dfrac{1}{14}$
- $\dfrac{25}{14}$
- $\dfrac{9}{14}$
- $\dfrac{19}{14}$
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The Correct Option is D
Solution and Explanation
Given: $8f(x) + 6f\left(\dfrac{1}{x}\right) = x + 5$
Put $x = 1$, get $8f(1) + 6f(1) = 1 + 5 \Rightarrow 14f(1) = 6 \Rightarrow f(1) = \dfrac{3}{7}$
Differentiate both sides w.r.t. $x$ and substitute $x=1$ to find $\dfrac{d}{dx} (x^2 f(x))$
Result is $\dfrac{19}{14}$
Put $x = 1$, get $8f(1) + 6f(1) = 1 + 5 \Rightarrow 14f(1) = 6 \Rightarrow f(1) = \dfrac{3}{7}$
Differentiate both sides w.r.t. $x$ and substitute $x=1$ to find $\dfrac{d}{dx} (x^2 f(x))$
Result is $\dfrac{19}{14}$
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