# Example 12 - Chapter 6 Class 12 Application of Derivatives

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Example 12 Find intervals in which the function given by f (x) = sin 3x, x, 0, 2 is (a) increasing (b) decreasing. f = sin 3 where 0 , 2 Step 1 :- Finding f (x) f = sin 3 f = sin 3 f = cos 3 . 3 = cos 3 . 3 = 3. cos 3 Step 2: Putting f = 0 3 cos 3 = 0 cos 3 = 0 We know that cos = 0 When = 2 & 3 2 3 = 2 & 3 = 3 2 = 2 3 & = 3 2 3 = 6 & = 2 Since = 6 0 , 2 & = 2 0, 2 both values of are valid Step 3: Plotting point Since 0 , 2 we start number line from 0 & end at 2 Point = 6 divide the interval 0 , 2 into two disjoint intervals 0 , 6 and 6 , 2 Step 4: Checking sign of f f = 3. cos 3 Case 1 In 0 , 6 0< < 6 3 0<3 < 3 6 0<3 < 2 So when 0 , 6 , then 3 0 , 2 And we know that cos >0 for 0 , 2 cos 3x >0 for 3x 0 , 2 cos 3x >0 for x 0 , 6 3 cos 3x >0 for x 0 , 6 ( )>0 for x 0 , 6 Since f (x) 0 for 0 , 6 Thus, f(x) is increasing for 0 , 6 Case 2 Since 6 , 2 6 < < 2 3 6 <3 < 3 2 2 <3 < 3 2 So when 6 , 2 , then 3 2 , 3 2 We know that, cos <0 for 2 , 3 2 cos 3 <0 for 3 2 , 3 2 cos 3 <0 for 6 , 2 3 cos 3 <0 for 6 , 2 f (x) <0 for 6 , 2 Since f (x) 0 for 6 , 2 Thus, f(x) is decreasing for 6 , 2 Thus, f(x) is increasing for , & f(x) is strictly decreasing for ,

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Chapter 6 Class 12 Application of Derivatives

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.