Class 12 Applications Of Derivatives · Very Long Answer5 Marksabsolute_extrema
Class 12 Maths · Applications Of Derivatives · Very Long Answer #104
Find the absolute maximum and minimum of f(x)=cos²x+sin x on [0,π].
✅ Step-by-Step Solution
Final Answer: Maximum 5/4 at π/6,5π/6; minimum 1 at 0,π/2,π.
- Continuity on [0,π] ensures extrema; f′=cos x(1−2sin x).
- Interior critical inputs are π/6,π/2,5π/6.
- At π/6 and 5π/6, f=3/4+1/2=5/4.
- At π/2, f=1; both endpoints 0 and π also give f=1.
- Comparing the values, the absolute maximum is 5/4 at π/6,5π/6 and the absolute minimum is 1 at 0,π/2,π.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Continuity on [0,π] ensures extrema; f′=cos x(1−2sin x).
- [1 mark] Interior critical inputs are π/6,π/2,5π/6.
- [1 mark] At π/6 and 5π/6, f=3/4+1/2=5/4.
- [1 mark] At π/2, f=1; both endpoints 0 and π also give f=1.
- [1 mark] Comparing the values, the absolute maximum is 5/4 at π/6,5π/6 and the absolute minimum is 1 at 0,π/2,π.
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