Paragraph: Let _ 1 :[0, ) R , _ 2 :[0, ) R , f:[0, ) R and g:[0, ) R be functions such that f(0)=g(0)=0, _ 1 (x)=e^ -x +x, x 0, _ 2 (x)=x^ 2 -2 x-2 e^ -x +2, x 0, f(x)= _ -x ^ x (|t|-t^ 2 ) e^ -t^ 2 d t, x>0 and g(x)= _ 0 ^ x^ 2 t e^ -t d t, x>0 Question: Which of the following statements is TRUE ?
D. g x ≤ 2 3 x 3 - 2 5 x 5 + 1 7 x 7 for all x ∈ 0 , 1 2
(A) Given ψ 1 x = e - x + x ,    x ≥ 0 ψ ' 1 x = - e - x + 1 ,    x ≥ 0 ψ ' 1 x is increasing function. ψ 1 x > ψ 1 0   ∀   x ≥ 0 ψ 1 x ≥ 1 So, e - x + x < 1 for x ∈ ( 0 , ∞ ) is incorrect. LHS is increasing and unbounded function. (B) Given ψ 2 x = x 2 - 2 x - 2 e - x + 2 ,    x ≥ 0 ψ ' 2 x = 2 x - 2 + 2 e - x ,    x ≥ 0 ψ ' 2 x = 2 ψ 1 x - 2   ≥ 0 ,    x ≥ 0 So, ψ 2 x is increasing function. x 2 - 2 x - 2 e - x + 2 < 1 for x ∈ ( 0 , ∞ ) is incorrect because LHS → ∞ when x → ∞ (C) f x = 2 ∫ 0 x t - t 2 e - t 2 d t f x = - e - t 2 0 x - 2 ∫ 0 x t 2 e - t 2 d t = 1 - e - x 2 - 2 ∫ 0 x t 2 1 - t 2 + t 4 2 ! ⋯ . . = f x - 1 + e - x 2 + 2 x 3 3 - 2 x 5 5 < 0 in 0 , 1 2 g x = ∫ 0 x 2 t e - 1 d t Put t = z 2 g x = ∫ 0 x 2 z 2 e - z 2 d z = ∫ 0 x 2 z 2 1 - z 2 + z 4 2 ! … . . d z g x = 2 x 3 3 - 2 x 5 5 + 2 x 7 2 . 7 - 2 · x 9 9 . 6 - … … g x - 2 x 3 3 + 2 x 5 5 - 2 x 7 2 . 7 ≤ 0 = 1 - e - x 2 - 2 x 3 3 + 2 x 5 5 - 2 x 7 2 + 2 x 9 6 … … … Now f x + g x = 1 - e - x 2 ⇒ f x = 1 - e - x 2 - g x ⇒ f x ≤ 1 - e - x 2 - 2 3 x 3 - 2 3 x 5 (D) g x = ∫ 0 x 2 t e - t d t ⇒ g x ≤ ∫ 0 x 2 t 1 - t + t 2 2 d t ⇒ g x ≤ 2 3 x 3 - 2 5 x 5 + 1 7 x 7 g x ≥ ∫ 0 x 2 t 1 - t d t ⇒ g x ≥ 2 3 x 3 - 2 5 x 5
Related: Mathematics — Application of Derivatives · All PYQ Banks