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  1. Chapter 6 Class 12 Application of Derivatives
  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‘ฅ1)<๐‘“(๐‘ฅ2) for ๐‘ฅ1 < ๐‘ฅ2 where ๐‘ฅ1 , ๐‘ฅ2 โˆˆ [๐‘Ž , ๐‘] Proof Let ๐‘ฅ1 , ๐‘ฅ2 be two numbers in the interval [๐‘Ž , ๐‘] i.e. ๐‘ฅ1 , ๐‘ฅ2 โˆˆ [๐‘Ž , ๐‘] & ๐‘ฅ1 < ๐‘ฅ2 Let us consider the interval [๐‘ฅ1 ," " ๐‘ฅ2] f is continuous & differentiable in [๐‘ฅ1 ," " ๐‘ฅ2] as f is continuous & differentiable in [๐‘Ž , ๐‘] 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‘ฅ2) โˆ’ ๐‘“(๐‘ฅ1))/(๐‘ฅ2 โˆ’ ๐‘ฅ1 ) Given that fโ€™(๐‘ฅ)>0 for all ๐‘ฅ โˆˆ (๐‘Ž , ๐‘) So, fโ€™(๐‘)>0 for all โˆˆ (๐‘ฅ1 ,๐‘ฅ2) (๐‘“(๐‘ฅ2) โˆ’ ๐‘“(๐‘ฅ1))/(๐‘ฅ2 โˆ’ ๐‘ฅ1 )>0 ๐‘“(๐‘ฅ2)โˆ’๐‘“(๐‘ฅ1)>0 โˆด ๐‘“(๐‘ฅ2)>๐‘“(๐‘ฅ1) So, for any two points ๐‘ฅ1 , ๐‘ฅ2 in interval [๐‘Ž , ๐‘] Where ๐‘ฅ2> ๐‘ฅ1 ๐‘“(๐‘ฅ2)> ๐‘“(๐‘ฅ1) So, f increasing in the interval [๐’‚ , ๐’ƒ]

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.