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

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 as the curve does not intersect X-axis i.e, f ' ( x ) ≠ 0 when x ∈ ( 0 , 1 ) (IV) Is false. f ' ' ( x ) ≠ 0 as tangent to the curve of f ' ( x ) is not parallel to X-axis. 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