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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 Column 2 Column

Options

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

Correct answer

B. (II) (ii) (Q)

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 (I) f 1 f e 2 0 , so true. (II) f ' 1 f ' e 0 , so true. (III) Graph of f ' x , so (III) is false. (IV) Is false. 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 , e 2 so true As f " x 0 ∀ x > 0 therefore, R is false, S is true. Al

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