Class 12 Applications Of Derivatives · Very Long Answer5 Markslocal_extrema
Class 12 Maths · Applications Of Derivatives · Very Long Answer #103
Find all local extrema of f(x)=sin x−cos x on 0<x<2π, including their values.
✅ Step-by-Step Solution
Final Answer: Local maximum √2 at 3π/4; local minimum −√2 at 7π/4.
- f′=cos x+sin x.
- Solving f′=0 in (0,2π) gives x=3π/4 and 7π/4.
- f″=−sin x+cos x=−f(x). At 3π/4 it is −√2<0, giving a local maximum.
- At 7π/4, f″=√2>0, giving a local minimum.
- The local maximum is f(3π/4)=√2; the local minimum is f(7π/4)=−√2.
📋 CBSE Stepwise Marking Scheme
- [1 mark] f′=cos x+sin x.
- [1 mark] Solving f′=0 in (0,2π) gives x=3π/4 and 7π/4.
- [1 mark] f″=−sin x+cos x=−f(x). At 3π/4 it is −√2<0, giving a local maximum.
- [1 mark] At 7π/4, f″=√2>0, giving a local minimum.
- [1 mark] The local maximum is f(3π/4)=√2; the local minimum is f(7π/4)=−√2.
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