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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

Misc 1
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Misc 2 Important

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important You are here

Misc 17 Important

Misc 18 Important

Misc. 19

Misc 20 Important

Misc 21 Important

Misc 22

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.