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
- Aψ 1 x ≤ 1 for all x > 0
- Bψ 2 x ≤ 0 , for all x > 0
- Cf x ≥ 1 - e - x 2 - 2 3 x 3 + 2 5 x 5 , for all x ∈ 0 , 1 2
- Dg x ≤ 2 3 x 3 - 2 5 x 5 + 1 7 x 7 for all x ∈ 0 , 1 2
Correct answer
D. g x ≤ 2 3 x 3 - 2 5 x 5 + 1 7 x 7 for all x ∈ 0 , 1 2
Step-by-step solution
(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