NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q3
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 3
Find increasing/decreasing intervals of f(x)=(4sin x−2x−x cos x)/(2+cos x).
💡 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: Increasing on (−π/2+2nπ,π/2+2nπ); decreasing on (π/2+2nπ,3π/2+2nπ), n∈Z.
- Rewrite f=4sin x/(2+cos x)−x. Differentiate to get f′=cos x(4−cos x)/(2+cos x)².
- Since 4−cos x and the denominator are positive, the derivative sign equals the sign of cos x.
- Use the positive and negative half-periods of cosine to obtain the intervals; continuity allows adjoining each stationary endpoint.
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