Ex 6.3,28 (MCQ) - Chapter 6 Class 12 Application of Derivatives

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Ex 6.3, 28 For all real values of x, the minimum value of (1 − 𝑥 + 𝑥2)/(1 + 𝑥 + 𝑥2) is (A) 0 (B) 1 (C) 3 (D) 1/3Let 𝑓(𝑥)=(1 − 𝑥 + 𝑥2)/(1 + 𝑥 + 𝑥2) Finding 𝒇′(𝒙) 𝑓(𝑥)=(1 − 𝑥 + 𝑥2)/(1 + 𝑥 + 𝑥2) 𝑓^′(𝑥) =((1 − 𝑥 + 𝑥^2 )^′ (1 + 𝑥 + 𝑥^2 ) − (1 − 𝑥 + 𝑥^2 ) (1 + 𝑥 + 𝑥^2 )^′)/(1 + 𝑥 + 𝑥^2 )^2 𝑓′(𝑥)=(−1 − 𝑥 − 𝑥^2 + 2𝑥 + 2𝑥^2+ 2𝑥^3 − (1 − 𝑥 + 𝑥^2+ 2𝑥 − 2𝑥^2 + 2𝑥^3 ))/(1 + 𝑥 + 𝑥^2 )^2 𝑓(𝑥)=(−1 + 𝑥 + 𝑥^2 + 2𝑥^3 − (1 + 𝑥 − 𝑥^2 + 2𝑥^3 ))/(1 + 𝑥 + 𝑥^2 )^2 𝑓′(𝑥)=(−2 + 2𝑥^2)/(1 + 𝑥 + 𝑥^2 )^2 Putting 𝒇^′ (𝒙)=𝟎 (−2 + 2𝑥^2)/(1 + 𝑥 + 𝑥^2 )^2 =0 2𝑥^2−2=0 2𝑥^2=2 𝑥^2=1 𝑥=±1 Hence, x = 1 or x = –1 are the critical points Finding value of 𝒇(𝒙) at critical points Hence, minimum value of f(x) is 1/3. So, (D) is the correct answer

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo