Class 12 Applications Of Derivatives · Long Answer3 Marksmonotonicity

Class 12 Maths · Applications Of Derivatives · Long Answer #53

Find maximal intervals of strict increase and decrease of f(x)=sin3x−cos3x for 0<x<π.

✅ Step-by-Step Solution

Final Answer: Increasing: (0,π/4] and [7π/12,11π/12]; decreasing: [π/4,7π/12] and [11π/12,π).

f′=3(cos3x+sin3x).
The roots in (0,π) are π/4,7π/12,11π/12.
The derivative signs across these roots are +,−,+,−. Each finite stationary endpoint may be included in either adjacent monotonicity interval.
Hence the increasing intervals are (0,π/4] and [7π/12,11π/12]; the decreasing intervals are [π/4,7π/12] and [11π/12,π).

[0.5 marks] f′=3(cos3x+sin3x).
[1 mark] The roots in (0,π) are π/4,7π/12,11π/12.
[1 mark] The derivative signs across these roots are +,−,+,−. Each finite stationary endpoint may be included in either adjacent monotonicity interval.
[0.5 marks] Hence the increasing intervals are (0,π/4] and [7π/12,11π/12]; the decreasing intervals are [π/4,7π/12] and [11π/12,π).

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