Ex 6.5,3 - Chapter 6 Class 12 Application of Derivatives - Part 24

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Ex 6.5,3 - Chapter 6 Class 12 Application of Derivatives - Part 25

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Ex 6.5,3 - Chapter 6 Class 12 Application of Derivatives - Part 26 Ex 6.5,3 - Chapter 6 Class 12 Application of Derivatives - Part 27

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

Transcript

Ex 6.5, 3 Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (vii) g (๐‘ฅ) = 1/(๐‘ฅ^2 + 2)Finding gโ€™(๐’™) gโ€™(๐‘ฅ)=๐‘‘/๐‘‘๐‘ฅ (1/(๐‘ฅ^2 + 2)) gโ€™(๐‘ฅ)=(๐‘‘(๐‘ฅ^2 + 2)^(โˆ’1))/๐‘‘๐‘ฅ gโ€™(๐‘ฅ)=โˆ’1(๐‘ฅ^2+2)^(โˆ’1โˆ’1) ร— (2๐‘ฅ+0) gโ€™(๐‘ฅ)=โˆ’2๐‘ฅ(๐‘ฅ^2+2)^(โˆ’2) gโ€ฒ(๐‘ฅ)=( โˆ’2๐‘ฅ )/(๐‘ฅ^2 + 2)^2 Putting gโ€™(๐’™)=๐ŸŽ ( โˆ’2๐‘ฅ )/(๐‘ฅ^2+2)^2 =0 โ€“2๐‘ฅ=0 ร—(๐‘ฅ^2+2)^2 โ€“2๐‘ฅ=0 ๐‘ฅ=0 Finding gโ€™โ€™(๐’™) gโ€™(๐‘ฅ)=(โˆ’2๐‘ฅ)/(๐‘ฅ^2 + 2)^2 gโ€™โ€™(๐‘ฅ)=(๐‘‘(โˆ’2๐‘ฅ)/๐‘‘๐‘ฅ . ใ€– (๐‘ฅ^2 + 2)ใ€—^2 โˆ’ (๐‘‘(๐‘ฅ^2 + 2)^2)/๐‘‘๐‘ฅ . (โˆ’2๐‘ฅ))/((๐‘ฅ^2 + 2)^2 )^2 Using quotient rule as (๐‘ข/๐‘ฃ)^โ€ฒ=(๐‘ข^โ€ฒ ๐‘ฃ โˆ’ ๐‘ฃ^โ€ฒ ๐‘ข)/๐‘ฃ^2 =(โˆ’2 (๐‘ฅ^2 + 2)^2โˆ’2 (๐‘ฅ^2 + 2)^(2โˆ’1).๐‘‘(๐‘ฅ^2 + 2)/๐‘‘๐‘ฅ . (โˆ’2๐‘ฅ))/((๐‘ฅ^2 + 2)^2 )^2 =(โˆ’2 (๐‘ฅ^2 + 2)^2โˆ’2 (๐‘ฅ^2 + 2)(2๐‘ฅ + 0) (โˆ’2๐‘ฅ))/(๐‘ฅ^2 + 2)^4 =(โˆ’2 (๐‘ฅ^2 + 2)^2โˆ’2 (๐‘ฅ^2 + 2)(2๐‘ฅ) (โˆ’2๐‘ฅ))/(๐‘ฅ^(2 )+ 2)^4 =(โˆ’2 (๐‘ฅ^2 + 2)^2+ 8๐‘ฅ^2 (๐‘ฅ^2 + 2))/(๐‘ฅ^(2 )+ 2)^4 =(โˆ’2 (๐‘ฅ^2 + 2)[(๐‘ฅ^(2 )+ 2) โˆ’ 4๐‘ฅ^2 ])/(๐‘ฅ^2 + 2)^4 =(โˆ’2 (๐‘ฅ^2 + 2)(โˆ’3๐‘ฅ^2 + 2))/(๐‘ฅ^2 + 2)^4 =(โˆ’2(โˆ’3๐‘ฅ^2 + 2))/(๐‘ฅ^(2 )+ 2)^3 Putting x = 0 in gโ€™โ€™(x) gโ€™โ€™(0)=(โˆ’2(โˆ’3(0) + 2))/(0^2 + 2)^3 =(โˆ’2(0 + 2))/(2)^3 =(โˆ’4)/8=(โˆ’1)/2 Hence gโ€™โ€™(๐‘ฅ)<0 when ๐‘ฅ = 0 โˆด ๐‘ฅ = 0 is point of local maxima Thus, g(๐‘ฅ) is maximum at ๐’™ = 0 Maximum value of g(๐’™) at x = 0 g(๐‘ฅ)=1/(๐‘ฅ^(2 )+ 2) g(0)=1/(0^2 + 2) = 1/2 Maximum value is ๐Ÿ/๐Ÿ

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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.