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 ?
C. For every x > 0 , there exists a β ∈ 0 , x such that ψ 2 x = 2 x ψ 1 β - 1
(A) Given f x = 2 ∫ 0 x t - t 2 e - t 2 d t ; x > 0 g x = ∫ 0 x 2 t e - 1 d t ; x > 0 put t = u 2 ∴ g x = 2 ∫ 0 x u 2 e - u 2 d u = 2 ∫ 0 x t 2 e - t 2 d t now, f x + g x = ∫ 0 x 2 t e - t 2 d t = 1 - e - x 2 ⇒ f ( ln 3 ) + g ( ln 3 ) = 1 - e - ln 3 = 2 3 (B) for α ∈ ( 1 , x ) , ψ 1 ( x ) = 1 + α x ⇒ e - x + x - 1 - α x = 0 ⇒ e - x - 1 = x ( α - 1 ) Which is not possible because LHS < 0 & RHS > 0 (C) ψ 2 x = 2 x ψ 1 β - 1 ψ ' 2 ( x ) = 2 ψ 1 ( x ) - 2 from LMVT, ψ 2 ( x ) - ψ 2 ( 0 ) x - 0 = ψ 2 ' β for atleast one β ∈ ( 0 , x ) ⇒ ψ 2 ( x ) = 2 x ψ 1 ( β ) - 1 (D) Given f x = ∫ - x x t - t 2 e - t 2 d t ,    x > 0 f ' x = 2 x - x 2 e - x 2 f ' x = 2 x 1 - x e - x 2 f ' x ≥ 0   for   x ∈ 0 , 1 So, increasing in this interval.   and   f ' x ≤ 0   for   x ∈ [ 1 , ∞ ) So, decreasing in this interval.
Related: Mathematics — Application of Derivatives · All PYQ Banks