Example 12 - Chapter 6 Class 12 Application of Derivatives

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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 ,