Class 12 Applications Of Derivatives · Very Long Answer5 Marksabsolute_extrema
Class 12 Maths · Applications Of Derivatives · Very Long Answer #66
Find the absolute maximum and minimum of f(x)=sin²x−cos x on [0,π].
✅ Step-by-Step Solution
Final Answer: Maximum 5/4 at 2π/3; minimum −1 at 0.
- The function is continuous on [0,π], so compare endpoints and all interior critical inputs.
- f′=sin x(2cos x+1); the only interior zero is x=2π/3.
- Endpoint values are f(0)=−1 and f(π)=1.
- At x=2π/3, f=3/4+1/2=5/4.
- Thus the absolute maximum is 5/4 at 2π/3 and the absolute minimum is −1 at 0.
📋 CBSE Stepwise Marking Scheme
- [1 mark] The function is continuous on [0,π], so compare endpoints and all interior critical inputs.
- [1 mark] f′=sin x(2cos x+1); the only interior zero is x=2π/3.
- [1 mark] Endpoint values are f(0)=−1 and f(π)=1.
- [1 mark] At x=2π/3, f=3/4+1/2=5/4.
- [1 mark] Thus the absolute maximum is 5/4 at 2π/3 and the absolute minimum is −1 at 0.
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