Class 12 Applications Of Derivatives · Long Answer3 Marksmonotonicity
Class 12 Maths · Applications Of Derivatives · Long Answer #54
Find the maximal monotonicity intervals of f(x)=−3ln(1+x)+4ln(2+x)−4/(2+x).
✅ Step-by-Step Solution
Final Answer: Decreasing on (−1,0], increasing on [0,∞).
- The real domain is x>−1. Differentiate: f′=−3/(1+x)+4/(2+x)+4/(2+x)².
- A common denominator gives f′=x(x+4)/[(x+1)(x+2)²].
- The denominator and x+4 are positive on the domain. Thus f′<0 for −1<x<0 and f′>0 for x>0.
- The function strictly decreases on (−1,0] and increases on [0,∞).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The real domain is x>−1. Differentiate: f′=−3/(1+x)+4/(2+x)+4/(2+x)².
- [1 mark] A common denominator gives f′=x(x+4)/[(x+1)(x+2)²].
- [1 mark] The denominator and x+4 are positive on the domain. Thus f′<0 for −1<x<0 and f′>0 for x>0.
- [0.5 marks] The function strictly decreases on (−1,0] and increases on [0,∞).
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