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Let f be real-valued function defined on the interval (0,∞) by
f(x)=lnx+ ∫_0^x▒√(1+sint) dt.Then which of the following statement(s) is (are) true ?
(A) f^'' (x) exists for all xϵ(0,∞)
(B) f^' (x) exists for all xϵ(0,∞)and f^' is continous on (0,∞),but not differentiable on (0,∞)
(C) there exists α 1 such that |f^' (x)| |f(x)| for all xϵ(α,∞)
(D) there exists β 0 such that |f(x)|+|f'(x)| β for all xϵ(0,∞)
JEE Advanced/ IIT JEE 2010 Maths PYQ
This is a conceptual JEE Advanced level question combining
logarithmic function + definite integral + differentiability.

Questions like this are very dangerous in exam because they test:
• existence of derivative
• continuity of derivative
• second derivative behaviour
• growth comparison of f(x) and f'(x)

If you master this, many future JEE Advanced questions become easy.

📘 Question (IIT-JEE Level)

Let f be a real-valued function defined on (0,\infty) by
f(x)=\ln x+\int_{0}^{x}\sqrt{1+\sin t}\,dt

Then which of the following statement(s) is (are) true?

(A) f''(x) exists for all x\in(0,\infty)
(B) f'(x) exists for all x\in(0,\infty) and is continuous but not differentiable on (0,\infty)
(C) There exists \alpha1 such that |f'(x)||f(x)| for all x\in(\alpha,\infty)
(D) There exists \beta0 such that |f(x)|+|f'(x)|\le\beta for all x\in(0,\

🎯 Why this PYQ is important

✔ Fundamental theorem of calculus
✔ Differentiability of integral functions
✔ Behaviour near x=0
✔ Growth comparison
✔ Classic JEE Advanced concept

This type of question repeats in IIT-JEE in different forms.


🚀 In this video you’ll learn
• How to differentiate integral functions quickly
• Where second derivative fails
• How to analyse behaviour near 0
• Short exam-oriented approach

Try solving yourself before watching solution.

⸻

🔔 For serious JEE aspirants

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More topics coming:
AOD
Definite Integration
Limits
Differential Equation
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