NCERT Class 12 · Ch 6 Exercise 6.2 · Q7
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Exercise 6.2 Question 7
Prove log(1+x)−2x/(2+x) is increasing for x>−1.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: Strictly increasing on (−1,∞).
- Differentiate: y′=1/(1+x)−4/(2+x)²=x²/[(1+x)(2+x)²].
- The denominator is positive on the domain, so y′≥0 and vanishes only at x=0.
- On any nonempty subinterval the derivative is positive somewhere (indeed except possibly at 0), so the function is strictly increasing.
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