Examples

Example 1

Example 2

Example 3

Example 4 Important

Example 5

Example 6

Example 7

Example 8 Important

Example 9 Important

Example 10

Example 11 Important

Example 12

Example 13 Important You are here

Example 14

Example 15

Example 16 Important

Example 17

Example 18 Important

Example 19

Example 20 Important

Example 21 Important

Example 22

Example 23 Important

Example 24

Example 25 Important

Example 26 Important

Example 27

Example 28 Important

Example 29 Important

Example 30 Important

Example 31 Important

Example 32 Important

Example 33 Important

Example 34 Important

Example 35

Example 36 Important

Example 37

Question 1 Deleted for CBSE Board 2025 Exams

Question 2 Deleted for CBSE Board 2025 Exams

Question 3 Deleted for CBSE Board 2025 Exams

Question 4 Important Deleted for CBSE Board 2025 Exams

Question 5 Deleted for CBSE Board 2025 Exams

Question 6 Deleted for CBSE Board 2025 Exams

Question 7 Deleted for CBSE Board 2025 Exams

Question 8 Deleted for CBSE Board 2025 Exams

Question 9 Deleted for CBSE Board 2025 Exams

Question 10 Deleted for CBSE Board 2025 Exams

Question 11 Deleted for CBSE Board 2025 Exams

Question 12 Deleted for CBSE Board 2025 Exams

Question 13 Important Deleted for CBSE Board 2025 Exams

Question 14 Important Deleted for CBSE Board 2025 Exams

Last updated at April 16, 2024 by Teachoo

Example 13 Find the intervals in which the function f given by f (π₯)=sinβ‘π₯+cosβ‘π₯ , 0 β€ π₯ β€ 2π is strictly increasing or strictly decreasing.f(π₯) = sin π₯ + cos π₯ Finding fβ(π) fβ(π₯) = (π )/ππ₯ (sin π₯ + cos π₯) fβ(π₯) = π(sinβ‘π₯ )/ππ₯ + π(cosβ‘π₯ )/ππ₯ fβ(π₯) = "cos " π₯ + (βπ πππ₯) fβ(π) = πππβ‘π β πππβ‘π Putting fβ(π) = 0 cos π₯ β sin π₯ = 0 cos π = sin π β΄π₯=π /π ,ππ /π ππ 0" β€ " π₯ β€ 2π Plotting points So, points π₯=π/4 ,5π/4 divides interval into 3 disjoint intervals [0 , π/4), (π/4,5π/4), (5π/4 , 2π] Checking sign of π^β² (π) π^β² (π₯)" "=" cos " π₯" β sin " π₯ When π β [π , π /π) Let us find value of fβ(x) at any value of π₯ lies between 0, π/4 Thus, fβ(π) > 0 for π₯ β [0 , π/4) At π = 0 fβ(0) = cos 0 β sin 0 = 1 β 0 = 1 > 0 At π = π /π β (π , π /π) fβ(π/6) = cos π/6 β sin π/6 = β3/2 β 1/2 = (β3 β 1)/2 =(1.73 β 1)/2=0.73/2 > 0 When π β (π /π,ππ /π) As π/4 < x < 5π/4 Let us find value of fβ(x) at any value of π₯ lies between π/4, 5π/4 Thus, fβ(π) < 0 for π₯ β (π/4,5π/4) Let π = π /π β (π /π,ππ /π) fβ (π₯) = cos π₯ β sin π₯ fβ(π/2) = cos π/2 β sin π/2 = 0 β 1 = β 1 < 0 When π β (ππ /π , ππ ] As 5π/4 < π₯ β€ 2π Let us find value of fβ(x) at any value of π₯ lies between 5π/4, 2π At π = 2Ο fβ(π₯) = cos π₯ β sin π₯ fβ(2π) = cos 2π β sin 2π = 1β0 = 1 > 0 Hence, fβ(x) > 0 for π₯ β (5π/4 , 2π] Thus, f is strictly increasing in intervals [π , π /π)& (ππ /π , ππ ] f is strictly increasing in intervals (π /π , ππ /π)