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Last updated at April 16, 2024 by Teachoo

Misc 13 Let f be a function defined on [a, b] such that fâ (đĽ) > 0, for all đĽ â (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(đđ)<đ(đđ) for đđ < đđ where đđ , đđ â [đ , đ] Proof Let đđ , đđ be two numbers in the interval [đ , đ] i.e. đĽ1 , đĽ2 â [đ , đ] And, đđ < đđ In Interval [đđ ," " đđ] As f is defined everywhere, f is continuous & differentiable in [đĽ1 ," " đĽ2] By Mean value of theorem, There exists c in (đĽ1 ,đĽ2) i.e. c â (đĽ1 ," " đĽ2) such that fâ(c) =(đ(đđ) â đ(đđ))/(đđ â đđ ) Given that fâ(đĽ)>0 for all đĽ â (đ , đ) So, fâ(đ)>đ for all c â (đđ ,đđ) (đ(đđ) â đ(đđ))/(đđ â đđ )>đ đ(đĽ2)âđ(đĽ1)>0 So, we can write that For any two points đĽ1 , đĽ2 in interval [đ , đ] Where đđ> đđ đ(đđ)> đ(đđ) Thus, f increasing in the interval [đ , đ] Hence proved