Misc 16 - Let f be a function defined on [a, b], f'(x) > 0

Misc 16 - Chapter 6 Class 12 Application of Derivatives - Part 2
Misc 16 - Chapter 6 Class 12 Application of Derivatives - Part 3

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

Transcript

Misc 16 Let f be a function defined on [a, b] such that fโ€™ (๐‘ฅ) > 0, for all ๐‘ฅ โˆˆ (a, b). Then prove that f is an increasing function on (a, b).We have to prove that function is always increasing i.e. f(๐’™๐Ÿ)<๐’‡(๐’™๐Ÿ) for ๐’™๐Ÿ < ๐’™๐Ÿ where ๐’™๐Ÿ , ๐’™๐Ÿ โˆˆ [๐’‚ , ๐’ƒ] Proof Let ๐’™๐Ÿ , ๐’™๐Ÿ be two numbers in the interval [๐‘Ž , ๐‘] i.e. ๐‘ฅ1 , ๐‘ฅ2 โˆˆ [๐‘Ž , ๐‘] And, ๐’™๐Ÿ < ๐’™๐Ÿ In Interval [๐’™๐Ÿ ," " ๐’™๐Ÿ] As f is defined everywhere, f is continuous & differentiable in [๐‘ฅ1 ," " ๐‘ฅ2] By Mean value of theorem, There exists c in (๐‘ฅ1 ,๐‘ฅ2) i.e. c โˆˆ (๐‘ฅ1 ," " ๐‘ฅ2) such that fโ€™(c) =(๐’‡(๐’™๐Ÿ) โˆ’ ๐’‡(๐’™๐Ÿ))/(๐’™๐Ÿ โˆ’ ๐’™๐Ÿ ) Given that fโ€™(๐‘ฅ)>0 for all ๐‘ฅ โˆˆ (๐‘Ž , ๐‘) So, fโ€™(๐’„)>๐ŸŽ for all c โˆˆ (๐’™๐Ÿ ,๐’™๐Ÿ) (๐’‡(๐’™๐Ÿ) โˆ’ ๐’‡(๐’™๐Ÿ))/(๐’™๐Ÿ โˆ’ ๐’™๐Ÿ )>๐ŸŽ ๐‘“(๐‘ฅ2)โˆ’๐‘“(๐‘ฅ1)>0 So, we can write that For any two points ๐‘ฅ1 , ๐‘ฅ2 in interval [๐‘Ž , ๐‘] Where ๐’™๐Ÿ> ๐’™๐Ÿ ๐’‡(๐’™๐Ÿ)> ๐’‡(๐’™๐Ÿ) Thus, f increasing in the interval [๐’‚ , ๐’ƒ] Hence proved

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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.