Last updated at April 19, 2021 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(๐๐)<๐(๐๐) for ๐๐ < ๐๐ where ๐๐ , ๐๐ โ [๐ , ๐] Proof Let ๐๐ , ๐๐ be two numbers in the interval [๐ , ๐] i.e. ๐ฅ1 , ๐ฅ2 โ [๐ , ๐] And, ๐๐ < ๐๐ In Interval [๐๐ ," " ๐๐] As f is defined everywhere, f is continuous & differentiable in [๐ฅ1 ," " ๐ฅ2] By Mean value of theorem, There exists c in (๐ฅ1 ,๐ฅ2) i.e. c โ (๐ฅ1 ," " ๐ฅ2) such that fโ(c) =(๐(๐๐) โ ๐(๐๐))/(๐๐ โ ๐๐ ) Given that fโ(๐ฅ)>0 for all ๐ฅ โ (๐ , ๐) So, fโ(๐)>๐ for all c โ (๐๐ ,๐๐) (๐(๐๐) โ ๐(๐๐))/(๐๐ โ ๐๐ )>๐ ๐(๐ฅ2)โ๐(๐ฅ1)>0 So, we can write that For any two points ๐ฅ1 , ๐ฅ2 in interval [๐ , ๐] Where ๐๐> ๐๐ ๐(๐๐)> ๐(๐๐) Thus, f increasing in the interval [๐ , ๐] Hence proved

Miscellaneous

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Misc 16 Important You are here

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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 10 years. He provides courses for Maths and Science at Teachoo.