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Let f : [2, 4] → R be a differentiable function such that (x log_e x) f'(x) + (log_e x) f(x) + f(x) ≥ 1, x ∈ [2, 4] with f(2) = 1/2 and f(4) = 1/2. Consider the following two statements: (A) f(x) ≤ 1, for all x ∈ [2, 4] (B) f(x) ≥ 1/8, for all x ∈ [2, 4] Then,

1. Neither statement (A) nor statement (B) is true
2. Only statement (B) is true
3. Both the statements (A) and (B) are true
4. Only statement (A) is true

Correct answer: Both the statements (A) and (B) are true

Solution

Both statements are true because the given inequality ensures that the function f(x) remains bounded above by 1 and does not drop below 1/8 within the specified interval, as confirmed by the boundary conditions f(2) = 1/2 and f(4) = 1/2.

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