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Ex 6.2, 18 - Prove that f(x) = x3 - 3x2 + 3x - 100 is increasing - To show increasing/decreasing in whole domain

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  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.2,18 Prove that the function given by f (𝑥) = 𝑥3 – 3𝑥2 + 3𝑥 – 100 is increasing in R. f﷐𝑥﷯ = 𝑥3 – 3𝑥2 + 3𝑥 – 100 We need to show f﷐𝑥﷯ is strictly increasing on R i.e. we need to show f’﷐𝑥﷯ > 0 Finding f’﷐𝑥﷯ f’﷐𝑥﷯= 3x2 – 6x + 3 – 0 = 3﷐𝑥2−2𝑥+1﷯ = 3﷐﷐𝑥﷯2+﷐1﷯2−2﷐𝑥﷯﷐1﷯﷯ = 3﷐𝑥−1﷯2 f’﷐𝑥﷯ = 3﷐𝑥−1﷯2 We know that Square of any number is always ﷐+﷯ve ⇒ ﷐𝑥−1﷯2 > 0 ⇒ 3﷐𝑥−1﷯2>0 ⇒ f’﷐𝑥﷯ > 0 Hence f’﷐𝑥﷯ > 0 for any value of 𝑥 ∴ f﷐𝑥﷯ is strictly increasing on R

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