Ex 6.2, 11 - Prove f(x) = x2-x+1 is neither strictly increasing - To show increasing/decreasing in intervals

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Ex 6.2,11 Prove that the function f given by f (𝑥) = ﷐𝑥﷮2﷯ – 𝑥 + 1 is neither strictly increasing nor strictly decreasing on (– 1, 1). Let f﷐𝑥﷯ = 𝑥2 – 𝑥 + 1 𝑥 ∈ ﷐−1 , 1﷯ Step 1: Finding f’﷐𝑥﷯ f’﷐𝑥﷯ = 2𝑥 – 1 Step 2: Putting f’﷐𝑥﷯ = 0 2𝑥 – 1 = 0 2𝑥 = 1 𝑥 = ﷐1﷮2﷯ Since 𝑥 ∈ ﷐−1 , 1﷯ The point 𝑥 = ﷐1﷮2﷯ divide the intervals ﷐−1 , 1﷯ into two disjoint intervals. i.e. ﷐−1 , ﷐1﷮2﷯﷯ & ﷐﷐1﷮2﷯ , 1﷯ Hence f’﷐𝑥﷯ < 0 for 𝑥 ∈ ﷐−1 , ﷐1﷮2﷯﷯ & f’﷐𝑥﷯ > 0 for 𝑥 ∈ ﷐﷐1﷮2﷯, 1﷯ Hence f﷐𝑥﷯ is neither decreasing nor increasing on ﷐−𝟏 , 𝟏﷯. Hence Proved

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