Question

If $5f(x) + 3f \left( \frac{1}{x} \right) = x + 2$ and $y = x f(x)$, then $\frac{dy}{dx}$ at $x = 1$ is
Identify the correct option from the following:

• A. 14
• B. $\frac{7}{8}$
• C. 1
• D. 2

Hint:

For functional equations, substitute $x$ with $\frac{1}{x}$ to form a system of equations, solve for the function, then apply the product rule for differentiation.

Answer 1

The Correct Option is B

Step 1: Solve for $f(x)$
Given $5f(x) + 3f \left( \frac{1}{x} \right) = x + 2$, substitute $x$ with $\frac{1}{x}$: $5f \left( \frac{1}{x} \right) + 3f(x) = \frac{1}{x} + 2$. Solve the system: $5f(x) + 3f \left( \frac{1}{x} \right) = x + 2$ (1), $3f(x) + 5f \left( \frac{1}{x} \right) = \frac{1}{x} + 2$ (2). Multiply (1) by 5, (2) by 3: $25f(x) + 15f \left( \frac{1}{x} \right) = 5x + 10$, $9f(x) + 15f \left( \frac{1}{x} \right) = \frac{3}{x} + 6$. Subtract: $16f(x) = 5x - \frac{3}{x} + 4$, $f(x) = \frac{5x}{16} - \frac{3}{16x} + \frac{1}{4}$. Step 2: Compute $y = x f(x)$ and its derivative
$y = x \left( \frac{5x}{16} - \frac{3}{16x} + \frac{1}{4} \right) = \frac{5x^2}{16} - \frac{3}{16} + \frac{x}{4}$. Then $\frac{dy}{dx} = \frac{10x}{16} + \frac{1}{4} = \frac{5x}{8} + \frac{1}{4}$. Step 3: Evaluate at $x = 1$
$\frac{dy}{dx} = \frac{5(1)}{8} + \frac{1}{4} = \frac{5}{8} + \frac{2}{8} = \frac{7}{8}$, which matches option (2).