Consider the function f : (– ∞, ∞) → (– ∞, ∞) defined by Let g(x)= _ 0 ^ e^ x f^ (t) 1+t^…

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

Consider the function f : (– ∞, ∞) → (– ∞, ∞) defined by

Let \[g(x)=\int_{0}^{e^{x}} \frac{f^{\prime}(t)}{1+t^{2}} d t\] Which of the following is true?

1. g′(x) is positive on (– ∞, 0) and negative on (0, ∞)
2. g′(x) is negative on (– ∞, 0) and positive on (0, ∞)
3. g′(x) changes sign on both (– ∞, 0) and (0, ∞)
4. g′(x) does not change sign on (– ∞, ∞).