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Last updated at Jan. 7, 2020 by Teachoo

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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(๐ฅ1)<๐(๐ฅ2) for ๐ฅ1 < ๐ฅ2 where ๐ฅ1 , ๐ฅ2 โ [๐ , ๐] Proof Let ๐ฅ1 , ๐ฅ2 be two numbers in the interval [๐ , ๐] i.e. ๐ฅ1 , ๐ฅ2 โ [๐ , ๐] & ๐ฅ1 < ๐ฅ2 Let us consider the interval [๐ฅ1 ," " ๐ฅ2] f is continuous & differentiable in [๐ฅ1 ," " ๐ฅ2] as f is continuous & differentiable in [๐ , ๐] By Mean value of theorem, there exists c in (๐ฅ1 ,๐ฅ2) i.e. c โ (๐ฅ1 ," " ๐ฅ2) such that fโ(c) =(๐(๐ฅ2) โ ๐(๐ฅ1))/(๐ฅ2 โ ๐ฅ1 ) Given that fโ(๐ฅ)>0 for all ๐ฅ โ (๐ , ๐) So, fโ(๐)>0 for all โ (๐ฅ1 ,๐ฅ2) (๐(๐ฅ2) โ ๐(๐ฅ1))/(๐ฅ2 โ ๐ฅ1 )>0 ๐(๐ฅ2)โ๐(๐ฅ1)>0 โด ๐(๐ฅ2)>๐(๐ฅ1) So, for any two points ๐ฅ1 , ๐ฅ2 in interval [๐ , ๐] Where ๐ฅ2> ๐ฅ1 ๐(๐ฅ2)> ๐(๐ฅ1) So, f increasing in the interval [๐ , ๐]

Miscellaneous

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Misc 2 Important

Misc 3 Important

Misc 4

Misc 5 Important

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Misc 8 Important

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Misc 11 Important

Misc 12 Important

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Misc 16 Important You are here

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Misc 20 Important

Misc 21 Important

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Misc. 23 Important

Misc 24 Important

Chapter 6 Class 12 Application of Derivatives

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About the Author

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.