Ex 6.2,19 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.2,19 The interval in which 𝑦 = 𝑥2 𝑒^(–𝑥) is increasing is (A) (– ∞, ∞) (B) (– 2, 0) (C) (2, ∞) (D) (0, 2) Let f(𝑥) = 𝑥^2 𝑒^(−𝑥) Finding f’(𝒙) f’(𝑥) = (𝑥^2 𝑒^(−𝑥) )′ f’(𝑥) = (𝑥2)′ e –𝑥 + (𝑒^(−𝑥) )’ (𝑥2) f’(𝑥) = (2𝑥) e –𝑥 + (〖−𝑒〗^(−𝑥) ) (𝑥2) f’(𝑥) = 2𝑥 e-𝑥 – e-𝑥 𝑥2 f’(𝑥) = 𝑥 e –𝑥 (2−𝑥) Putting f’(𝒙)=𝟎 𝑥 e-𝑥 (2−𝑥) = 0 𝑥 (2−𝑥) = 0 So, 𝑥=0 & 𝑥 = 2 Plotting points on real line The points 𝑥 = 0 & 2 divide the real line into 3 disjoint intervals i.e (−∞ , 0) , (0 , 2) & (2 , ∞) (As e –𝑥 is always positive for all 𝑥 ∈ R) Hence, f(𝑥) is strictly increasing on (0 , 2) Therefore, correct answer is (𝐃)