Ex 6.3,28 (MCQ) - Chapter 6 Class 12 Application of Derivatives
Last updated at April 16, 2024 by Teachoo
Local maxima and minima
Last updated at April 16, 2024 by Teachoo
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