JEE Advanced2021MathematicsApplication of DerivativesActual
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)=x²-2 x-2 e^ -x +2, x 0 , f(x)= _ -x ^ x (|t|-t² ) e^ -t² d t, x>0 and g(x)= ₀^ x² t e^ -t d t, x>0 Question: Which of the following statements is TRUE ?
Options
- Af ln 3 + g ln 3 = 1 3
- BFor every x > 1 , there exists an α ∈ 1 , x such that ψ 1 x = 1 + α x
- CFor every x > 0 , there exists a β ∈ 0 , x such that ψ 2 x = 2 x ψ 1 β - 1
- Df is an increasing function on the interval 0 , 3 2
Correct answer
C. For every x > 0 , there exists a β ∈ 0 , x such that ψ 2 x = 2 x ψ 1 β - 1
Step-by-step solution
(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