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JEE Advanced2017MathematicsApplication of DerivativesActual

Let f x = x + log e ⁡ x - x log e ⁡ x , x ∈ 0 , ∞ • Column 1 contains information about zeros of f ( x ) , f ' ( x ) and f ' ' ( x ) • Column 2 contains information about the limiting behaviour of f ( x ) , f ' ( x ) and f ' ' ( x ) at infinity. • Column 3 contains information about increasing-decreasing nature of f ( x ) and f ' ( x ) Column 1 Col

Options

  1. A(III) (iv) (P)
  2. B(I) (ii) (R)
  3. C(II) (iii) (S)
  4. D(IV) (i) (S)

Correct answer

C. (II) (iii) (S)

Step-by-step solution

f x = x + ℓ n x - x ℓ n x , x > 0 f ' x = 1 + 1 x - ℓ n x - 1 f ' ' x = - 1 x 2 - 1 x = - x + 1 x 2 In Column 1, (I) f 1 f e 2 0 so true (II) f ' 1 f ' e 0 so true (III) f ' x is always positive for ( 0 , 1 ) so it cannot be zero. So (III) is false. (IV) Is false In Column 2 As lim x → ∞ ⁡ f x = lim x → ∞ ⁡ x 1 + ℓ n x x - ℓ n x = - ∞ ∴ (i) is false (ii) is true lim x → ∞ ⁡ f ' x = - ∞ ,so (iii) is true. lim x → ∞ ⁡ f ' ' x = 0 , so (iv) is true. (P) f ' x is positive in ( 0 , 1 ) ,so true. (Q) f ' x 0 ,for in e ,

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