Ex 6.2, 12 - Which functions are strictly decreasing on - Ex 6.2

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 Ex 6.2,12 - Chapter 6 Class 12 Application of Derivatives - Part 4 Ex 6.2,12 - Chapter 6 Class 12 Application of Derivatives - Part 5 Ex 6.2,12 - Chapter 6 Class 12 Application of Derivatives - Part 6 Ex 6.2,12 - Chapter 6 Class 12 Application of Derivatives - Part 7

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Ex 6.2, 12 Which of the following functions are strictly decreasing on (0 ,πœ‹/2)? (A) cos π‘₯f(π‘₯) = cos π‘₯ f’(𝒙) = – sin 𝒙 Since, sin π‘₯ > 0 for π‘₯ ∈ (0 , πœ‹/2) So, –sin 𝒙 < 0 for π‘₯ ∈ (0 , πœ‹/2) ∴ f’(π‘₯) < 0 for π‘₯ ∈ (0 , πœ‹/2) So, f is strictly decreasing in (0 , πœ‹/2) 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) Ex 6.2, 12 Which of the following functions are strictly decreasing on (0,πœ‹/2) ? (C) cos 3π‘₯ Let f(π‘₯) = cos 3π‘₯ Finding f’(𝒙) f’(π‘₯) = (cos⁑3π‘₯ )β€² f’(π‘₯) = –3 sin 3π‘₯ Let 3π‘₯ = ΞΈ ∴ f’(π‘₯) = –3 sin ΞΈ When 0 < x < πœ‹/2 , then 0 < ΞΈ < πŸ‘π…/𝟐 For 0 < ΞΈ < πŸ‘π…/𝟐 sin ΞΈ is positive for 0 < ΞΈ < πœ‹ sin ΞΈ is negative for 0 < ΞΈ < πŸ‘π…/𝟐 Thus, we can say that sin ΞΈ is neither positive nor negative for 0 < ΞΈ < πŸ‘π…/𝟐 Putting ΞΈ = 3x sin 3x is neither positive nor negative for 0 < 3x < πŸ‘π…/𝟐 βˆ’3 sin 3x is neither positive nor negative for 0 < 3x < 3πœ‹/2 f’(x) is neither positive nor negative for 0 < x < 𝝅/𝟐 Thus, we can write that f(x) is neither increasing nor decreasing for 𝒙 ∈ (𝟎 , 𝝅/𝟐) Ex 6.2, 12 Which of the following functions are strictly decreasing on (0,πœ‹/2) ? (D) tan π‘₯f(π‘₯) = tan π‘₯ f’(𝒙) = sec2 𝒙 As square of any number is always positive So, f’(π‘₯) > 0 for all values of π‘₯ ∴ f is strictly increasing on (0 , πœ‹/2)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.