Class 12 Applications Of Derivatives · Long Answer3 Marksmonotonicity
Class 12 Maths · Applications Of Derivatives · Long Answer #56
Show that f(x)=ln(1+x)−2x/(2+x) is strictly increasing throughout x>−1.
✅ Step-by-Step Solution
Final Answer: Strictly increasing on (−1,∞).
- For x>−1 both logarithm and denominator are defined. Differentiate: f′=1/(1+x)−4/(2+x)².
- Simplification gives f′=x²/[(x+1)(x+2)²].
- This is nonnegative on the domain and positive except at the single point 0. On every nontrivial interval the integral of f′ is positive.
- Hence f is strictly increasing on (−1,∞), despite f′(0)=0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For x>−1 both logarithm and denominator are defined. Differentiate: f′=1/(1+x)−4/(2+x)².
- [1 mark] Simplification gives f′=x²/[(x+1)(x+2)²].
- [1 mark] This is nonnegative on the domain and positive except at the single point 0. On every nontrivial interval the integral of f′ is positive.
- [0.5 marks] Hence f is strictly increasing on (−1,∞), despite f′(0)=0.
Master Chapter 6: Applications Of Derivatives
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Applications Of Derivatives.
← Previous Question
Find the maximal intervals of strict increase and decrease of…
Next Question →Prove f(x)=2x+arccot x+ln(√(1+x²)−x) is strictly increasing on R…
More Practice Questions in Applications Of Derivatives
Related Questions in Applications Of Derivatives
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximal open interval on which f(x)=x²e^(−x) has positive derivative.
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=x²−4x+6, identify the maximal open interval where f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
For x>0, at which x is f(x)=x^x stationary?
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum value of (1/x)^x for x>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum of [x(x−1)+1]^(1/3) on 0≤x≤1.
Class 12 Applications Of Derivatives · Mcq Single Correct
A trapezium area model is A(x)=(10+x)√(100−x²) cm² for 0≤x≤10. Find its maximum.