Last updated at April 19, 2021 by

Transcript

Misc 6 Find the intervals in which the function f given by f(π₯) = (4 sinβ‘γπ₯ β 2π₯ β π₯πππ π₯γ)/(2 + cosβ‘π₯ ) is (i) increasing (ii) decreasing.f(π₯) = (4 sinβ‘γπ₯ β 2π₯ β π₯πππ π₯γ)/(2 + cosβ‘π₯ ) Letβs consider the interval [π , ππ ] Finding fβ(π) f(π₯) = (4 sinβ‘γπ₯ β 2π₯ β π₯ πππ π₯γ)/(2 + cosβ‘π₯ ) f(π₯) = (4 sinβ‘γπ₯ β π₯(2 + cosβ‘π₯ )γ)/(2 + cosβ‘π₯ ) f(π₯) = (4 sinβ‘π₯)/(2 + πππ ) β π₯(2 + cosβ‘π₯ )/(2 + cosβ‘π₯ ) f(π) = (π πππβ‘π)/(π + πππβ‘π )βπ Differentiating w.r.t π₯ fβ(π₯) = π/ππ₯ ((4 sinβ‘π₯)/(2 + πππ π₯) β π₯) = π/ππ₯ ((4 sinβ‘π₯)/(2 + cosβ‘π₯ )) β π(π₯)/ππ₯ = π/ππ₯ ((4 sinβ‘π₯)/(2 + cosβ‘π₯ )) β 1 Using quotient rule as (π’/π£)^β² = (π’^β² π£ βπ£^β² π’)/π£^2 Using quotient rule as (π’/π£)^β² = (π’^β² π£ βπ£^β² π’)/π£^2 = (8 πππ π₯ β cos^2β‘π₯ β 4 cosβ‘π₯ )/(2 +γ cosγβ‘π₯ )^2 = (4 cosβ‘γπ₯ βγ cosγ^2β‘π₯ γ)/(2 + cosβ‘π₯ )^2 β΄ fβ(π) = πππβ‘γπ (π β πππβ‘π )γ/(π + πππβ‘π )^π Putting fβ(π) = 0 cosβ‘π₯(4 β cosβ‘π₯ )/(2 + πππ β‘π₯ )^2 = 0 Thus, Numerator is 0 cos π (πβπππβ‘π ) = 0 Thus, cos π = 0 Since 0 β€ π₯ β€ 2π So, values of π are π /π & ππ /π 4 β cos π₯ = 0 cos π₯ = 4 But β1" β€" cosβ‘π₯ β€ 1 So cos π₯ = 4 is not possible 4 β cos π₯ = 0 cos π₯ = 4 But β1" β€" cosβ‘π₯ β€ 1 So cos π₯ = 4 is not possible Plotting value of π on number line So, f(π₯) is strictly increasing on (0 , π/2) & (3π/2 , 2π) f(π₯) is strictly decreasing on (π/2 ,3π/2) But we need to find Increasing & Decreasing fβ(π₯) = πππ β‘γπ₯ (4 β πππ β‘π₯ )γ/(2 + πππ β‘π₯ )^2 Thus, f(π₯) is increasing on [π , π /π] & [ππ /π , ππ ] f(π₯) is decreasing on [π /π ,ππ /π] For x = 0 fβ(0) = 1/3 For x = π/2 fβ(π/2) = 0 For x = 3π/2 fβ(3π/2) = 0 For x = 2π fβ(2π) = 1/3

Miscellaneous

Misc 1 (a)
Deleted for CBSE Board 2022 Exams

Misc 1 (b) Important Deleted for CBSE Board 2022 Exams

Misc 2 Important

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6 Important You are here

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16

Misc 17 Important

Misc 18 Important

Misc. 19 (MCQ) Deleted for CBSE Board 2022 Exams

Misc 20 (MCQ) Important

Misc 21 (MCQ) Important

Misc 22 (MCQ)

Misc. 23 (MCQ) Important

Misc 24 (MCQ) Important

Chapter 6 Class 12 Application of Derivatives (Term 1)

Serial order wise

About the Author

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.