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﷯ 𝑢﷐1, 𕔴uc1﷯ 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﷯ 𝑢﷐1, 𕔴uc1﷯ Step 1: Finding f’﷐𝑥﷯ f﷐𝑥﷯ = 𝑥 + ﷐1﷮𝑥﷯ f’﷐𝑥﷯ = 1 – ﷐1﷮𝑥2﷯ f’﷐𝑥﷯ = ﷐𝑥2 − 1﷮𝑥2﷯ f’﷐𝑥﷯ = ﷐﷐1﷮𝑥2﷯﷯﷐𝑥−1﷯﷐𝑥+1﷯ Step 2: Putting f’﷐𝑥﷯ = 0 ﷐𝑥2 − 1﷮𝑥2﷯ = 0 𝑥2−1 = 0 ﷐𝑥−1﷯﷐𝑥+1﷯=0 So, 𝑥=1 & 𝑥=−1 Step 3: Plotting point on real line The point 𝑥 = –1 , 1 into three disjoint intervals i.e. ﷐−𕔴 −1﷯ ﷐−1 , 1﷯ & ﷐1 , 𕔴uc1﷯ Step 4: ∴ f﷐𝑥﷯ is strictly increasing on ﷐−𕔴 −𝟏﷯ & ﷐𝟏 , 𕔴uc1﷯ Hence proved