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Ex 6.2,12 - Chapter 6 Class 12 Application of Derivatives - Part 2

Ex 6.2,12 - Chapter 6 Class 12 Application of Derivatives - Part 3

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Ex 6.2, 12 Which of the following functions are strictly decreasing on (0,πœ‹/2) ? (B) cos 2π‘₯ Let f(π‘₯) = cos 2π‘₯ Finding f’(𝒙) f’(π‘₯) = (cos⁑2π‘₯ )β€² f’(π‘₯) = –2 sin 2π‘₯ Let 2π‘₯ = ΞΈ ∴ f’(π‘₯) = –2 sin ΞΈ When 0 < x < πœ‹/2 , then 0 < ΞΈ < πœ‹ Since sin ΞΈ > 0 for 0 < ΞΈ < πœ‹ Therefore, sin 2x > 0 for 0 < 2x < πœ‹ Multiplying βˆ’2 both sides βˆ’2 Γ— sin 2x < βˆ’2 Γ— 0 for 0 < 2x < πœ‹ βˆ’2 sin 2x < 0 for 0 < 2x < πœ‹ f’(x) < 0 for 0 < 2x < πœ‹ f’(x) < 0 for 0 < x < 𝝅/𝟐 Therefore, f(x) is strictly decreasing for 𝒙 ∈ (𝟎 , 𝝅/𝟐)

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