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Example 47 - Find intervals in which f(x) = 3/10x4 - 4/5x3 - Find intervals of increasing/decreasing

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Example 47 Find intervals in which the function given by f(𝑥) =﷐3﷮10﷯𝑥4 – ﷐4﷮5﷯﷐𝑥﷮3﷯– 3𝑥2 + ﷐36﷮5﷯𝑥 + 11 is (a) strictly increasing (b) strictly decreasing f﷐𝑥﷯ = ﷐3﷮10﷯𝑥4 – ﷐4﷮5﷯﷐𝑥﷮3﷯– 3𝑥2 + ﷐36﷮5﷯𝑥 + 11 Step 1: Finding f’﷐𝑥﷯ f’﷐𝑥﷯ = ﷐3﷮10﷯ × 4﷐𝑥﷮3﷯ – ﷐4﷮5﷯ × 3﷐𝑥﷮2﷯ – 3 × 2x + ﷐36﷮5﷯ + 0 f’﷐𝑥﷯ = ﷐12﷮10﷯﷐𝑥﷮3﷯– ﷐12﷮5﷯﷐𝑥﷮2﷯– 6x + ﷐36﷮5﷯ f’﷐𝑥﷯ = ﷐6﷮5﷯﷐𝑥﷮3﷯− ﷐12﷮5﷯﷐𝑥﷮2﷯– 6x + ﷐36﷮5﷯ f’﷐𝑥﷯ = 6﷐﷐﷐𝑥﷮3﷯﷮5﷯−﷐2﷐𝑥﷮2﷯﷮5﷯−𝑥+﷐6﷮5﷯﷯ f’﷐𝑥﷯ = 6﷐﷐﷐𝑥﷮3﷯ − 2﷐𝑥﷮2﷯− 5𝑥 + 6﷮5﷯﷯ = ﷐6﷮5﷯ ﷐﷐𝑥﷮3﷯−2﷐𝑥﷮2﷯−5𝑥+6﷯ = ﷐6﷮5﷯﷐𝑥−1﷯﷐𝑥2−𝑥−6﷯ = ﷐6﷮5﷯ ﷐𝑥−1﷯﷐𝑥2−3𝑥+2𝑥−6﷯ = ﷐6﷮5﷯ ﷐𝑥−1﷯﷐𝑥﷐𝑥−3﷯+2﷐𝑥−3﷯﷯ = ﷐6﷮5﷯ ﷐𝑥−1﷯﷐𝑥+2﷯﷐𝑥−3﷯ Hence f’﷐𝑥﷯ = ﷐6﷮5﷯﷐𝑥−1﷯﷐𝑥+2﷯﷐𝑥−3﷯ Step 2: Putting f’﷐𝑥﷯=0 ﷐6﷮5﷯﷐𝑥−1﷯﷐𝑥+2﷯﷐𝑥−3﷯ = 0 ﷐𝑥−1﷯﷐𝑥+2﷯﷐𝑥−3﷯ = 0 Hence x = –2 , 1 & 3 Step 3: Plotting point on real line Thus, we get four disjoint intervals i.e. ﷐−𕔴−2﷯ ,﷐−2, 1﷯ ,﷐1 , 3﷯, ﷐3 , 𕔴uc1﷯ ⇒ f﷐𝑥﷯ is strictly decreasing on the interval 𝑥 ∈﷐−𕔴−𝟐﷯& ﷐𝟏 , 𝟑﷯ f﷐𝑥﷯ is strictly increasing on the interval 𝑥 ∈﷐−𝟐,𝟏﷯ & ﷐𝟑 , 𕔴uc1﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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