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JEE Main202311 Apr 2023Morning ShiftMathematicsApplication of DerivativesActual

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,

Options

  1. ANeither statement ( A ) nor statement ( B ) is true
  2. BOnly statement ( B ) is true
  3. CBoth the statements ( A ) and ( B )   are true
  4. DOnly statement ( A ) is true

Correct answer

C. Both the statements ( A ) and ( B )   are true

Step-by-step solution

Given, Domain and range of function, f : 2 , 4 → ℝ x log e x f ' x + log e x f x + f x ≥ 1 ,   x ∈ 2 , 4 ⇒ x log e x d d x f x + log e x f x d d x x + x f x d d x log e x ≥ 1 ⇒ d d x x ln x f x ≥ 1 ⇒ d d x x ln x f x ≥ d d x x ⇒ d d x x ln x f x - x ≥ 0 Hence, h ( x ) = x ln x f x - x is a increasing function, ∴ h ( x ) ≥ h ( 2 ) , x ∈ [ 2 , 4 ] ⇒ x ln x × f x - x ≥ 2 ln 2 × f 2 - 2 ⇒ x ln

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