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

To determine the truthfulness of the two statements regarding the function \( f(x) \), we start by analyzing the given information.

The function \( f: [2, 4] \to \mathbb{R} \) is differentiable, and it satisfies the inequality:

\((x \log x) f'(x) + (\log x) f(x) \geq 1\) for \( x \in [2, 4] \).

We also know the boundary values of the function: \( f(2) = \frac{1}{2} \) and \( f(4) = \frac{1}{4} \).

Let's first explore statement (A):

• \((A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4]\)

To check the validity of this statement, assume \( f(x) \geq 1 \). However, at the boundary points \( x = 2 \) and \( x = 4 \), we find:

• \( f(2) = \frac{1}{2} \), which is less than 1.
• \( f(4) = \frac{1}{4} \), which is also less than 1.

Thus, statement (A) is false since it contradicts the given boundary conditions of \( f(x) \).

Next, let's analyze statement (B):

• \((B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4]\)

If \( f(x) \leq \frac{1}{8} \) over the entire interval, this would contradict the given boundary conditions since:

• \( f(2) = \frac{1}{2} \) and \( f(4) = \frac{1}{4} \), both values are greater than \(\frac{1}{8}\).

Hence, statement (B) is also false.

Given both statements (A) and (B) lead to contradictions concerning the boundary values, neither can be true. However, according to the correct answer provided, both statements are claimed to be true.

This paradox suggests a misunderstanding or misinterpretation in the presentation of the information, because logically, neither statement should hold if strictly interpreted. Based on a typically correct analysis, the conclusion should correctly identify that, mathematically speaking, the provided answer is incorrect under consistency with boundary values.