Question

Let \( f: [2, 4] \to \mathbb{R} \) be a differentiable function such that \( (x \log x) f'(x) + (\log x) f(x) \geq 1 \), \( x \in [2, 4] \) with \( f(2) = \frac{1}{2} \) and \( f(4) = \frac{1}{4} \). Consider the following two statements:
\( (A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4] \)
\( (B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4] \) Then,

• A. Only statement (B) is true
• B. Only statement (A) is true
• C. Neither statement (A) nor statement (B) is true
• D. Both the statements (A) and (B) are true

Hint:

When dealing with inequalities involving logarithmic and exponential functions, applying differentiation and simplifying terms can reveal useful properties of the function.

Answer 1

The Correct Option is D

We are given that \( x \cdot \log x \cdot f'(x) + \log x \cdot f(x) \geq 1 \) for \( x \in [2, 4] \), and also that \( f(2) = \frac{1}{2} \) and \( f(4) = \frac{1}{4} \).
First, we differentiate the given inequality: \[ \frac{d}{dx} \left( x \cdot \log x \cdot f(x) \right) \geq 0 \] This leads to: \[ \frac{d}{dx} \left( f(x) \cdot \log x \right) \geq 0 \] Now, simplifying the derivatives: \[ \frac{d}{dx} \left( (f(x) \cdot \log x) \right) \Rightarrow f'(x) \cdot \log x + f(x) \cdot \frac{1}{x} \geq 0 \] This ensures that \( f(x) \) is increasing and positive in the interval \( [2, 4] \).
Next, we define a new function \( g(x) = \ln(x) f(x) - x \). We then find that \( g(x) \) is increasing in the interval \( [2, 4] \).
Now, we solve for the behavior of \( f(x) \) using the boundaries of the interval \( [2, 4] \): \[ f(2) = \frac{1}{2}, \quad f(4) = \frac{1}{4} \] We compute the bounds and find that the value of \( f(x) \) falls between the values of \( \frac{1}{2} \) and \( \frac{1}{8} \), which leads to the conclusion that both statements (A) and (B) are true.