Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12   1. Chapter 6 Class 12 Application of Derivatives
2. Serial order wise
3. Miscellaneous

Transcript

Misc 16 Let f be a function defined on [a, b] such that f’ (𝑥) > 0, for all 𝑥 ∈ (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(𝑥1)<𝑓(𝑥2) for 𝑥1 < 𝑥2 where 𝑥1 , 𝑥2 ∈ [𝑎 , 𝑏] Proof Let 𝑥1 , 𝑥2 be two numbers in the interval [𝑎 , 𝑏] i.e. 𝑥1 , 𝑥2 ∈ [𝑎 , 𝑏] & 𝑥1 < 𝑥2 Let us consider the interval [𝑥1 ," " 𝑥2] f is continuous & differentiable in [𝑥1 ," " 𝑥2] as f is continuous & differentiable in [𝑎 , 𝑏] By Mean value of theorem, there exists c in (𝑥1 ,𝑥2) i.e. c ∈ (𝑥1 ," " 𝑥2) such that f’(c) =(𝑓(𝑥2) − 𝑓(𝑥1))/(𝑥2 − 𝑥1 ) Given that f’(𝑥)>0 for all 𝑥 ∈ (𝑎 , 𝑏) So, f’(𝑐)>0 for all ∈ (𝑥1 ,𝑥2) (𝑓(𝑥2) − 𝑓(𝑥1))/(𝑥2 − 𝑥1 )>0 𝑓(𝑥2)−𝑓(𝑥1)>0 ∴ 𝑓(𝑥2)>𝑓(𝑥1) So, for any two points 𝑥1 , 𝑥2 in interval [𝑎 , 𝑏] Where 𝑥2> 𝑥1 𝑓(𝑥2)> 𝑓(𝑥1) So, f increasing in the interval [𝒂 , 𝒃]

Miscellaneous 