Class 12 Applications Of Derivatives · Short Answer2 Marksmonotonicity
Class 12 Maths · Applications Of Derivatives · Short Answer #45
Prove f(x)=16sin x/(4+cos x)−x is strictly decreasing on (π/2,π).
✅ Step-by-Step Solution
Final Answer: Strictly decreasing on (π/2,π).
- The quotient rule gives f′=16(4cos x+1)/(4+cos x)²−1.
- Simplify to cos x(56−cos x)/(4+cos x)². Here cos x<0 while 56−cos x>0 and the denominator is positive.
- Thus f′<0 throughout (π/2,π), proving strict decrease.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The quotient rule gives f′=16(4cos x+1)/(4+cos x)²−1.
- [1 mark] Simplify to cos x(56−cos x)/(4+cos x)². Here cos x<0 while 56−cos x>0 and the denominator is positive.
- [0.5 marks] Thus f′<0 throughout (π/2,π), proving strict decrease.
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