Let f be a real-valued function defined on the interval (0, ∞) by f(x)= x+ _ 0 ^ x 1+ t d…

Mathematics · JEE Advanced · NTA Exams — Limit, Continuity and Differentiability

1. f ′′(x) exists for all x ∈ (0, ∞)
2. f ′(x) exists for all x ∈ (0, ∞) and f ′ is continuous on (0, ∞), but not differentiable on (0, ∞)
3. there exists α > 1 such that |f ′(x)| < |f (x)| for all
   x ∈ (α, ∞)
4. there exists β > 0 such that |f(x)| + |f ′(x)| ≤ β for all x ∈ (0, ∞)