Example 32 - Find local maximum and local minimum values - Local maxima and minima

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Example 32 (Method 1) Find local maximum and local minimum values of the function f given by f ( )=3 4 + 4 3 12 2 + 12 f ( )=3 4 + 4 3 12 2 + 12 Step 1: Finding f ( ) f ( )= 3 4 + 4 3 12 2 + 12 f ( )=12 3 +12 2 24 + 0 f ( )=12 3 +12 2 24 f ( )=12 3 + 2 2 f =12 2 +2 2 f =12 +2 1 +2 f =12 1 +2 Step 2: Putting f =0 12 1 +2 =0 1 +2 =0 So, =0 , 1 & 2 are critical points Step 3: Step 4: Finding local minimum and maximum value Local maximum value of f ( ) at =0 f (0)=3 0 4 + 4 0 3 12 0 2 + 12 = 0 + 0 0 + 12 = 12 Local minimum value of f ( ) at =1 f (1)=3 1 4 + 4 1 3 12 1 2 + 12 = 3 + 4 12 + 12= 7 Local Minimum value of f ( ) at = 2 f ( 2)=3 2 4 + 4 2 3 12 2 2 + 12 = 48 32 48 + 12 = 20 Example 32(Method 2) Find local maximum and local minimum values of the function f given by f ( )=3 4 + 4 3 12 2 + 12 f ( )=3 4 + 4 3 12 2 + 12 Step 1: Finding f ( ) f ( )= 3 4 + 4 3 12 2 + 12 f ( )=12 3 +12 2 24 + 0 f ( )=12 3+12 3 24 f ( )=12 3 + 2 2 f =12 2 + 2 f =12 2 +2 2 f =12 +2 1 +2 f =12 1 +2 Step 2: Putting f =0 12 1 +2 =0 1 +2 =0 So, =0, =1,& = 1 Step 3: Finding f f =12 3 + 2 2 f = 12 3 + 2 2 f =12 3 2 +2 2 At = f 0 =12 3 0 2 +2 0 2 = 32 0+0 2 = 64 < 0 Since f <0 at =0 is point of local maxima Thus, f is maximum at =0 At = f 1 =12 3 1 2 +2 1 2 = 12 3+2 2 = 36 > 0 Since f >0 at =1 is point of local minima Thus, f is minimum at =1 At = f 2 =12 3 2 2 +2 2 2 = 12 3 4 4 2 = 72 > 0 Since f >0 at = 2 is point of local minima Thus, f is minimum at = 2 Step 4: Finding local minimum and maximum value Local maximum value of f ( ) at =0 f (0)=3 0 4 + 4 0 3 12 0 2 + 12 = 0 + 0 0 + 12 = 12 Local minimum value of f ( ) at =1 f (1)=3 1 4 + 4 1 3 12 1 2 + 12 = 3 + 4 12 + 12 = 7 Local Minimum value of f ( ) at = 2 f ( 2)=3 2 4 + 4 2 3 12 2 2 + 12 = 48 32 48 + 12 = 20

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.