Misc 16 - Let f be a function defined on [a, b], f'(x) > 0 - Miscellaneous

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  1. Chapter 6 Class 12 Application of Derivatives
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Misc 16 Let f be a function defined on [a, b] such that f ( ) > 0, for all (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 1 < 2 for 1 < 2 where 1 , 2 , Proof: Let 1 , 2 be two numbers in the interval , i.e. 1 , 2 , & 1 < 2 Let us consider the interval 1 , 2 f is continuous & differentiable in 1 , 2 as f is continuous & differentiable in , By Mean value of theorem, there exists c in 1 , 2 i.e. c 1 , 2 such that f = 2 1 2 1 Given that f >0 for all , So, f >0 for all 1 , 2 2 1 2 1 >0 2 1 >0 2 > 1 So, for two points 1 , 2 in interval , Where 2> 1 2 > 1 So, f increasing in the interval ,

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.