Question

Let $ f(x) + 2f\left( \frac{1}{x} \right) = x^2 + 5 $ and $ 2g(x) - 3g\left( \frac{1}{2} \right) = x, \, x>0. \, \text{If} \, \alpha = \int_{1}^{2} f(x) \, dx, \, \beta = \int_{1}^{2} g(x) \, dx, \text{ then the value of } 9\alpha + \beta \text{ is:}$

• A. 1
• B. 0
• C. 10
• D. 11

Hint:

In problems with integrals and functions, break down the system of equations and substitute values to simplify the expression for integration.

Answer 1

The Correct Option is D

We are given: \[ f(x) + 2f\left( \frac{1}{x} \right) = x^2 + 5 \] Substitute \( x = \frac{1}{x} \) into the equation: \[ f\left( \frac{1}{x} \right) + 2f(x) = \frac{1}{x^2} + 5 \] Now solve these two equations for \( f(x) \). First, we rewrite the system of equations:
1. \( f(x) + 2f\left( \frac{1}{x} \right) = x^2 + 5 \)
2. \( f\left( \frac{1}{x} \right) + 2f(x) = \frac{1}{x^2} + 5 \) Multiply the first equation by 2 and subtract from the second equation: \[ 2f(x) + 4f\left( \frac{1}{x} \right) = 2x^2 + 10 \] Subtract the second equation: \[ \left( 2f(x) + 4f\left( \frac{1}{x} \right) \right) - \left( f\left( \frac{1}{x} \right) + 2f(x) \right) = 2x^2 + 10 - \left( \frac{1}{x^2} + 5 \right) \] This simplifies to: \[ 3f\left( \frac{1}{x} \right) = 2x^2 - \frac{1}{x^2} + 5 \] From this, we can now solve for \( f(x) \). Next, for \( g(x) \), we are given: \[ 2g(x) - 3g\left( \frac{1}{2} \right) = x \] This simplifies to: \[ g(x) = \frac{x + 3g\left( \frac{1}{2} \right)}{2} \] For \( g(x) \), we substitute \( g\left( \frac{1}{2} \right) = \frac{1}{2} \) (after solving) and calculate \( \beta \) using the integral. Finally, using the formulas for \( \alpha \) and \( \beta \), we compute \( 9\alpha + \beta \).
Thus, the correct value of \( 9\alpha + \beta = 11 \).
Therefore, the correct answer is \( 11 \).

Answer 2

Step 1: The given equation is: 

\[ f(x) + 2f\left( \frac{1}{x} \right) = x^2 + 5 \]

Step 2: Substitute and simplify:

\[ f\left( \frac{1}{x} \right) + 2f(x) = \frac{1}{x^2} + 5 \] Solving for \( f(x) \): \[ f(x) = \frac{2}{3x^2} - \frac{x^2}{3} + \frac{5}{3} \]

Step 3: Find the integral for \( \alpha \):

\[ \alpha = \int_1^2 \left( \frac{2}{3x^2} - \frac{x^2}{3} + \frac{5}{3} \right) dx \] Simplifying the integral: \[ \left( \frac{2}{3x^2} - \frac{x^2}{3} + \frac{5}{3} \right) = -\frac{1}{3} + \frac{8}{9} + \frac{10}{3} + \frac{2}{3} + \frac{1}{9} - \frac{5}{3} \] Thus: \[ \alpha = \frac{2}{3} - \frac{7}{9} = \frac{11}{9} \]

Step 4: Equation for \( g(x) \):

\[ 2g(x) - 3g\left( \frac{1}{2} \right) = x \] For \( g\left( \frac{1}{2} \right) \), we have: \[ g\left( \frac{1}{2} \right) = -\frac{1}{2} \] Solving for \( g(x) \): \[ g(x) = \frac{x}{2} - \frac{3}{4} \]

Step 5: Find the integral for \( \beta \):

\[ \beta = \int_1^2 \left( \frac{x}{2} - \frac{3}{4} \right) dx \] Simplifying the integral: \[ \left( \frac{x^2}{4} - \frac{3x}{4} \right) = 1 - \frac{3}{4} - \frac{1}{4} + \frac{3}{4} = 0 \]

Step 6: Final calculation:

\[ 9\alpha + \beta = 11 \]