Ex 6.2,15 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.2, 15 Let I be any interval disjoint from [–1, 1]. Prove that the function f given by 𝑓(𝑥) = 𝑥 + 1/𝑥 is strictly increasing on I. I is any interval disjoint from [–1, 1] Hence I = (−∞, −1)∪(1, ∞) f(𝑥) = 𝑥 + 1/𝑥 We need to show f(𝑥) is strictly increasing on I i.e. we need to show f’(𝑥) > 0 for 𝑥 ∈(−∞, −1)∪(1, ∞) Finding f’(𝒙) f(𝑥) = 𝑥 + 1/𝑥 f’(𝑥) = 1 – 1/𝑥2 f’(𝑥) = (𝑥2 − 1)/𝑥2 Putting f’(𝒙) = 0 (𝑥2 − 1)/𝑥2 = 0 𝑥2−1 = 0 (𝑥−1)(𝑥+1)=0 So, 𝑥=1 & 𝑥=−1 Plotting point on real line The point 𝑥 = –1 , 1 into three disjoint intervals i.e. (−∞ , −1) (−1 , 1) & (1 , ∞) ∴ f(x) is strictly increasing on (−∞ , −𝟏) & (𝟏 , ∞) Hence proved