# Misc 16

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 16 Let f be a function defined on [a, b] such that fโ (๐ฅ) > 0, for all ๐ฅ โ (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๏ท๐ฅ1๏ทฏ<๐๏ท๐ฅ2๏ทฏ for ๐ฅ1 < ๐ฅ2 where ๐ฅ1 , ๐ฅ2 โ ๏ท๐ , ๐๏ทฏ Proof: Let ๐ฅ1 , ๐ฅ2 be two numbers in the interval ๏ท๐ , ๐๏ทฏ i.e. ๐ฅ1 , ๐ฅ2 โ ๏ท๐ , ๐๏ทฏ & ๐ฅ1 < ๐ฅ2 Let us consider the interval ๏ท๐ฅ1 , ๐ฅ2๏ทฏ f is continuous & differentiable in ๏ท๐ฅ1 , ๐ฅ2๏ทฏ as f is continuous & differentiable in ๏ท๐ , ๐๏ทฏ By Mean value of theorem, there exists c in ๏ท๐ฅ1 ,๐ฅ2๏ทฏ i.e. c โ ๏ท๐ฅ1 , ๐ฅ2๏ทฏ such that fโ๏ท๐๏ทฏ=๏ท๐๏ท๐ฅ2๏ทฏ โ ๐๏ท๐ฅ1๏ทฏ๏ทฎ๐ฅ2 โ ๐ฅ1 ๏ทฏ Given that fโ๏ท๐ฅ๏ทฏ>0 for all ๐ฅ โ ๏ท๐ , ๐๏ทฏ So, fโ๏ท๐๏ทฏ>0 for all โ ๏ท๐ฅ1 ,๐ฅ2๏ทฏ ๏ท๐๏ท๐ฅ2๏ทฏโ๐๏ท๐ฅ1๏ทฏ๏ทฎ๐ฅ2 โ๐ฅ1 ๏ทฏ>0 ๐๏ท๐ฅ2๏ทฏโ๐๏ท๐ฅ1๏ทฏ>0 โด ๐๏ท๐ฅ2๏ทฏ>๐๏ท๐ฅ1๏ทฏ So, for two points ๐ฅ1 , ๐ฅ2 in interval ๏ท๐ , ๐๏ทฏ Where ๐ฅ2> ๐ฅ1 ๐๏ท๐ฅ2๏ทฏ> ๐๏ท๐ฅ1๏ทฏ So, f increasing in the interval ๏ท๐ , ๐๏ทฏ

Chapter 6 Class 12 Application of Derivatives

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.