Last updated at Aug. 19, 2021 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(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

Misc 1 (a)
Deleted for CBSE Board 2022 Exams

Misc 1 (b) Important Deleted for CBSE Board 2022 Exams

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 You are here

Misc 17 Important

Misc 18 Important

Misc. 19 (MCQ) Deleted for CBSE Board 2022 Exams

Misc 20 (MCQ) Important

Misc 21 (MCQ) Important

Misc 22 (MCQ)

Misc. 23 (MCQ) Important

Misc 24 (MCQ) Important

Chapter 6 Class 12 Application of Derivatives (Term 1)

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