Ex 6.2,8 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.2,8 Find the values of 𝑥 for which y = [𝑥(𝑥 – 2)]2 is an increasing function 𝑦 =﷐﷐𝑥﷐𝑥−2﷯﷯﷮2﷯ Step 1: Finding ﷐𝑑𝑦﷮𝑑𝑥﷯ 𝑦 =﷐﷐𝑥﷐𝑥−2﷯﷯﷮2﷯ 𝑦 =﷐﷐﷐𝑥﷮2﷯−2𝑥﷯﷮2﷯ 𝑦 =﷐﷐𝑥﷯﷮4﷯+﷐﷐2𝑥﷯﷮2﷯−2﷐﷐𝑥﷮2﷯﷯﷐2𝑥﷯ 𝑦 = ﷐𝑥﷮4﷯+4﷐𝑥﷮2﷯−4﷐𝑥﷮3﷯ Diff. w.r.t 𝑥 ﷐𝑑𝑦﷮𝑑𝑥﷯=﷐𝑑﷐﷐𝑥﷮4﷯+4﷐𝑥﷮2﷯−4﷐𝑥﷮3﷯﷯﷮𝑑𝑥﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4﷐𝑥﷮3﷯+8𝑥−12﷐𝑥﷮2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4𝑥﷐﷐𝑥﷮2﷯+2−3𝑥﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4𝑥﷐﷐𝑥﷮2﷯−3𝑥+2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4𝑥﷐﷐𝑥﷮2﷯−2𝑥−𝑥+2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4𝑥﷐𝑥﷐𝑥−2﷯−1﷐𝑥−2﷯﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4𝑥﷐﷐𝑥−1﷯﷐𝑥−2﷯﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯=4𝑥﷐𝑥−1﷯﷐𝑥−2﷯ Step 2: Putting ﷐𝑑𝑦﷮𝑑𝑥﷯=0 4𝑥﷐𝑥−1﷯﷐𝑥−2﷯=0 So, 𝑥=0 , 𝑥=1 & 𝑥=2 Step 3: Plotting points on real line The points 𝑥 = 0 , 1 and 2 divide the real line into 4 disjoint intervals i.e. ﷐−𕔴0﷯ , ﷐0 , 1﷯ , ﷐1 , 2﷯ & ﷐2 ,𕔴uc1﷯ Step 4: Thus the function y = ﷐𝑥﷐﷐𝑥−1﷯﷮2﷯﷯ is strictly increasing for 0 <𝑥<1 and 𝑥>2 Thus the function y = ﷐𝑥﷐﷐𝑥−1﷯﷮2﷯﷯ is strictly increasing for 0 <𝒙<𝟏 and 𝒙>𝟐