Ā  Ex 6.3, 3 (vii) - For g(x) = 1/(x^2 + 2), find local maxima & minima - Ex 6.3

part 2 - Ex 6.3, 3 (vii) - Ex 6.3 - Serial order wise - Chapter 6 Class 12 Application of Derivatives
part 3 - Ex 6.3, 3 (vii) - Ex 6.3 - Serial order wise - Chapter 6 Class 12 Application of Derivatives
part 4 - Ex 6.3, 3 (vii) - Ex 6.3 - Serial order wise - Chapter 6 Class 12 Application of Derivatives

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Ex 6.3, 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 (š‘„) = 1/(š‘„^2 + 2)Finding g’(š’™) g’(š‘„)=š‘‘/š‘‘š‘„ (1/(š‘„^2 + 2)) g’(š‘„)=(š‘‘(š‘„^2 + 2)^(āˆ’1))/š‘‘š‘„ g’(š‘„)=āˆ’1(š‘„^2+2)^(āˆ’1āˆ’1) Ɨ (2š‘„+0) g’(š‘„)=āˆ’2š‘„(š‘„^2+2)^(āˆ’2) g′(š‘„)=( āˆ’2š‘„ )/(š‘„^2 + 2)^2 Putting g’(š’™)=šŸŽ ( āˆ’2š‘„ )/(š‘„^2+2)^2 =0 –2š‘„=0 Ɨ(š‘„^2+2)^2 –2š‘„=0 š‘„=0 Finding g’’(š’™) g’(š‘„)=(āˆ’2š‘„)/(š‘„^2 + 2)^2 g’’(š‘„)=(š‘‘(āˆ’2š‘„)/š‘‘š‘„ . 怖 (š‘„^2 + 2)怗^2 āˆ’ (š‘‘(š‘„^2 + 2)^2)/š‘‘š‘„ . (āˆ’2š‘„))/((š‘„^2 + 2)^2 )^2 =(āˆ’2 (š‘„^2 + 2)^2āˆ’2 (š‘„^2 + 2)^(2āˆ’1).š‘‘(š‘„^2 + 2)/š‘‘š‘„ . (āˆ’2š‘„))/((š‘„^2 + 2)^2 )^2 =(āˆ’2 (š‘„^2 + 2)^2āˆ’2 (š‘„^2 + 2)(2š‘„ + 0) (āˆ’2š‘„))/(š‘„^2 + 2)^4 =(āˆ’2 (š‘„^2 + 2)^2āˆ’2 (š‘„^2 + 2)(2š‘„) (āˆ’2š‘„))/(š‘„^(2 )+ 2)^4 =(āˆ’2 (š‘„^2 + 2)^2+ 8š‘„^2 (š‘„^2 + 2))/(š‘„^(2 )+ 2)^4 =(āˆ’2 (š‘„^2 + 2)[(š‘„^(2 )+ 2) āˆ’ 4š‘„^2 ])/(š‘„^2 + 2)^4 =(āˆ’2 (š‘„^2 + 2)(āˆ’3š‘„^2 + 2))/(š‘„^2 + 2)^4 =(āˆ’2(āˆ’3š‘„^2 + 2))/(š‘„^(2 )+ 2)^3 Putting x = 0 in g’’(x) g’’(0)=(āˆ’2(āˆ’3(0) + 2))/(0^2 + 2)^3 =(āˆ’2(0 + 2))/(2)^3 =(āˆ’4)/8=(āˆ’1)/2 Hence g’’(š‘„)<0 when š‘„ = 0 ∓ š‘„ = 0 is point of local maxima Thus, g(š‘„) is maximum at š’™ = 0 Maximum value of g(š’™) at x = 0 g(š‘„)=1/(š‘„^(2 )+ 2) g(0)=1/(0^2 + 2) = 1/2 Maximum value is šŸ/šŸ

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo