Last updated at April 19, 2021 by

Transcript

Example 12 Find intervals in which the function given by f (x) = sin 3x, x, β [0, π/2] is (a) increasing (b) decreasing. f(π₯) = sin 3π₯ where π₯ β [0 ,π/2] Finding fβ(x) fβ(π₯) = π(sinβ‘3π₯ )/ππ₯ fβ(π₯) = cos 3π₯ Γ 3 fβ(π) = 3. cos 3π Putting fβ(π) = 0 3 cos 3π₯ = 0 cos 3π₯ = 0 We know that cos ΞΈ = 0 When ΞΈ = π/2 & 3π/2 So, for cos 3π = 0 3π₯ = π/2 & 3π₯ = 3π/2 π₯ = π/(2 Γ3) & π₯ = 3π/(2 Γ 3) π = π /π & π = π /π Since π₯ = π/6 β [π ,π /π] & π₯ = π/2 β [π,π /π] β΄ Both values of π₯ are valid Plotting points on number line So, point π₯ = π/6 divides the interval into two disjoint intervals [0 ,π/6) and (π/6, π/2] Checking sign of fβ(π) fβ(π₯) = 3. cos 3π₯ Case 1: For π β (π ,π /π) 0<π₯<π/6 3 Γ 0<3π₯<3π/6 π<ππ<π /π So when π₯ β (0 ,π/6), then 3π₯ β (0 , π/2) We know that cos π½>π for π½ β (π , π /π) cos 3x >0 for 3x β (0 , π/2) cos 3x >0 for x β (0 , π/6) 3 cos 3x >0 for x β (0 , π/6) πβ²(π)>π for x β (0 , π/6) Since fβ(0) = 3 and fβ(π /π) = 0 Therefore, fβ(x) β₯ 0 for π₯ β [0 , π/6] Thus, f(x) is increasing for π₯ β [0 , π/6] Case 2: For π β (π /π, π /π) π/6<π₯<π/2 3 Γ π/6<3π₯<3π/2 π /π<ππ<ππ /π So when π₯ β(π/6 , π/2), then 3π₯ β (π/2 , 3π/2) We know that, cos π<0 for π β (π/2 , 3π/2) cos 3π₯<0 for 3π₯ β (π/2 , 3π/2) cos 3π₯<0 for π₯ β (π/6 , π/2) 3 cos 3π₯<0 for π₯ β (π/6 , π/2) fβ(x) <π for π₯ β (π/6 , π/2) Since fβ(π /π) = 0 and fβ(π /π) = 0 Therefore fβ(x) β€ 0 for π₯ β [π/6,π/2] Thus, f(x) is decreasing for π₯ β [π/6,π/2] (As cos π is negative in 2nd and 3rd quadrant) Thus, f(x) is increasing for π β [π , π /π] & f(x) is decreasing for π β [π /π , π /π]

Examples

Example 1
Deleted for CBSE Board 2022 Exams

Example 2 Deleted for CBSE Board 2022 Exams

Example 3 Deleted for CBSE Board 2022 Exams

Example 4 Important Deleted for CBSE Board 2022 Exams

Example 5 Deleted for CBSE Board 2022 Exams

Example 6 Deleted for CBSE Board 2022 Exams

Example 7

Example 8 Important

Example 9 Important

Example 10

Example 11 Important

Example 12 You are here

Example 13 Important

Example 14

Example 15

Example 16

Example 17 Important

Example 18

Example 19

Example 20

Example 21 Deleted for CBSE Board 2022 Exams

Example 22 Deleted for CBSE Board 2022 Exams

Example 23 Deleted for CBSE Board 2022 Exams

Example 24 Deleted for CBSE Board 2022 Exams

Example 25 Deleted for CBSE Board 2022 Exams

Example 26

Example 27

Example 28 Important

Example 29

Example 30 Important

Example 31

Example 32 Important

Example 33 Important

Example 34

Example 35 Important

Example 36

Example 37 Important

Example 38 Important

Example 39

Example 40 Important

Example 41 Important

Example 42 Important Deleted for CBSE Board 2022 Exams

Example 43 Important Deleted for CBSE Board 2022 Exams

Example 44 Important Deleted for CBSE Board 2022 Exams

Example 45 Important Deleted for CBSE Board 2022 Exams

Example 46 Important

Example 47 Important

Example 48 Important

Example 49 Deleted for CBSE Board 2022 Exams

Example 50 Important

Example 51

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.