Misc 2 - Show that f(x) = log x/x has maximum at x = e - Local maxima and minima

Slide10.JPG
Slide11.JPG Slide12.JPG

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

Transcript

Misc 2 Show that the function given by f(x) = log﷮𝑥﷯﷮𝑥﷯ has maximum at x = e. Let f 𝑥﷯ = log﷮𝑥﷯﷮𝑥﷯ Step 1: Finding f’ 𝑥﷯ f 𝑥﷯ = 𝑙𝑜𝑔﷮𝑥﷯ f’ 𝑥﷯ = 𝑑﷮𝑑𝑥﷯ log﷮𝑥﷯﷮𝑥﷯﷯ f’ 𝑥﷯ = 𝑑 log﷮𝑥﷯﷯﷮𝑑𝑥﷯ . 𝑥 − 𝑑 𝑥﷯﷮𝑑𝑥﷯ . log 𝑥﷮𝑥2﷯ f’ 𝑥﷯ = 1﷮2﷯ × 𝑥 − log﷮𝑥﷯﷮𝑥2﷯ f’ 𝑥﷯ = 1 − log﷮𝑥﷯﷮𝑥2﷯ Step 2: Putting f’ 𝑥﷯ = 0 1 − log﷮𝑥﷯﷮𝑥2﷯=0 1 – log 𝑥 = 0 log 𝑥 = 1 𝑥 = e Step 3: Finding f’’ 𝑥﷯ f’ 𝑥﷯ = 1 − log﷮𝑥﷯﷮𝑥2﷯ Diff w.r.t. 𝑥 f’’ 𝑥﷯ = 𝑑﷮𝑑𝑥﷯ 1 − log﷮𝑥﷯﷮𝑥2﷯﷯ f’’ 𝑥﷯ = 𝑑 1 − log﷮𝑥﷯﷯﷮𝑑𝑥﷯ . 𝑥2− 𝑑 𝑥2﷯﷮𝑑𝑥﷯ . 1 − log﷮𝑥﷯﷯﷮ 𝑥﷮2﷯﷯﷮2﷯﷯ = 0 − 1﷮𝑥﷯﷯ . 𝑥2 − 2𝑥 1 − log﷮𝑥﷯﷯﷮𝑥4﷯ = −1﷮𝑥﷯ × 𝑥2 − 2𝑥 1 − log﷮𝑥﷯﷯﷮ 𝑥﷮4﷯﷯ = −𝑥 − 2𝑥 1 − log﷮𝑥﷯﷯﷮ 𝑥﷮4﷯﷯ = −𝑥 1 + 2 1 − log﷮𝑥﷯﷯﷯﷮𝑥4﷯ = −𝑥 1 + 2 − 2 log﷮𝑥﷯﷯﷮ 𝑥﷮4﷯﷯ = −𝑥 3 − 2 log﷮𝑥﷯﷯﷮𝑥4﷯ f’’ 𝑥﷯ = − 3 − 2 log﷮𝑥﷯﷯﷮𝑥3﷯ Putting 𝑥 = e f’’ 𝑒﷯ = − 3 − 2 log﷮𝑒﷯﷯﷮𝑒3﷯ = − 3 − 2﷯﷮𝑒3﷯ = −1﷮𝑒3﷯ = – 1﷮𝑒3﷯﷯ < 0 ⇒ f’’ 𝑥﷯ < 0 at 𝑥 = e . ⇒ 𝑥 = e is point of maxima ⇒ f 𝑥﷯ is maximum at 𝒙 = e.

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail