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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Misc 1 Show that the function given by f(x) = log⁡𝑥/𝑥 is maximum at x = e.Let f(𝑥) = log⁡𝑥/𝑥 Finding f’(𝒙) f’(𝑥) = 𝑑/𝑑𝑥 (log⁡𝑥/𝑥) f’(𝑥) = (𝑑(log⁡𝑥 )/𝑑𝑥 " " . 𝑥 − 𝑑(𝑥)/𝑑𝑥 " . log " 𝑥)/𝑥2 f’(𝑥) = (1/𝑥 × 𝑥 − log⁡𝑥)/𝑥2 f’(𝑥) = (1 − log⁡𝑥)/𝑥2 Putting f’(𝒙) = 0 (1 − log⁡𝑥)/𝑥2=0 1 – log 𝑥 = 0 log 𝑥 = 1 𝒙 = e 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[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 Since f’’(𝑥) < 0 at 𝑥 = e . ∴ 𝑥 = e is point of maxima Hence, f(𝑥) is maximum at 𝒙 = e.

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.