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

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  1. Chapter 6 Class 12 Application of Derivatives
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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๏ท๐‘ฅ1๏ทฏ<๐‘“๏ท๐‘ฅ2๏ทฏ for ๐‘ฅ1 < ๐‘ฅ2 where ๐‘ฅ1 , ๐‘ฅ2 โˆˆ ๏ท๐‘Ž , ๐‘๏ทฏ Proof: Let ๐‘ฅ1 , ๐‘ฅ2 be two numbers in the interval ๏ท๐‘Ž , ๐‘๏ทฏ i.e. ๐‘ฅ1 , ๐‘ฅ2 โˆˆ ๏ท๐‘Ž , ๐‘๏ทฏ & ๐‘ฅ1 < ๐‘ฅ2 Let us consider the interval ๏ท๐‘ฅ1 , ๐‘ฅ2๏ทฏ f is continuous & differentiable in ๏ท๐‘ฅ1 , ๐‘ฅ2๏ทฏ as f is continuous & differentiable in ๏ท๐‘Ž , ๐‘๏ทฏ By Mean value of theorem, there exists c in ๏ท๐‘ฅ1 ,๐‘ฅ2๏ทฏ i.e. c โˆˆ ๏ท๐‘ฅ1 , ๐‘ฅ2๏ทฏ such that fโ€™๏ท๐‘๏ทฏ=๏ท๐‘“๏ท๐‘ฅ2๏ทฏ โˆ’ ๐‘“๏ท๐‘ฅ1๏ทฏ๏ทฎ๐‘ฅ2 โˆ’ ๐‘ฅ1 ๏ทฏ Given that fโ€™๏ท๐‘ฅ๏ทฏ>0 for all ๐‘ฅ โˆˆ ๏ท๐‘Ž , ๐‘๏ทฏ So, fโ€™๏ท๐‘๏ทฏ>0 for all โˆˆ ๏ท๐‘ฅ1 ,๐‘ฅ2๏ทฏ ๏ท๐‘“๏ท๐‘ฅ2๏ทฏโˆ’๐‘“๏ท๐‘ฅ1๏ทฏ๏ทฎ๐‘ฅ2 โˆ’๐‘ฅ1 ๏ทฏ>0 ๐‘“๏ท๐‘ฅ2๏ทฏโˆ’๐‘“๏ท๐‘ฅ1๏ทฏ>0 โˆด ๐‘“๏ท๐‘ฅ2๏ทฏ>๐‘“๏ท๐‘ฅ1๏ทฏ So, for two points ๐‘ฅ1 , ๐‘ฅ2 in interval ๏ท๐‘Ž , ๐‘๏ทฏ Where ๐‘ฅ2> ๐‘ฅ1 ๐‘“๏ท๐‘ฅ2๏ทฏ> ๐‘“๏ท๐‘ฅ1๏ทฏ So, f increasing in the interval ๏ท๐’‚ , ๐’ƒ๏ทฏ

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