Class 12 Applications Of Derivatives · Long Answer3 Marksmonotonicity
Class 12 Maths · Applications Of Derivatives · Long Answer #52
Find the maximal intervals of strict increase and decrease of f(x)=tan x−4x on 0<x<π/2.
✅ Step-by-Step Solution
Final Answer: Decreasing on (0,π/3], increasing on [π/3,π/2).
- f′=sec²x−4.
- Since cos x>0 here, f′=0 gives cos x=1/2, so x=π/3.
- For x<π/3, sec²x<4 and f′<0; for x>π/3, sec²x>4 and f′>0.
- Therefore f strictly decreases on (0,π/3] and increases on [π/3,π/2).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] f′=sec²x−4.
- [1 mark] Since cos x>0 here, f′=0 gives cos x=1/2, so x=π/3.
- [1 mark] For x<π/3, sec²x<4 and f′<0; for x>π/3, sec²x>4 and f′>0.
- [0.5 marks] Therefore f strictly decreases on (0,π/3] and increases on [π/3,π/2).
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