Ex 6.2, 5 - Find intervals f(x) = 2x3 - 3x2 - 36x + 7 - 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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Ex 6.2,5 Find the intervals in which the function f given by f (๐‘ฅ) = 2๐‘ฅ3 โ€“ 3๐‘ฅ2 โ€“ 36๐‘ฅ + 7 is (a) strictly increasing (b) strictly decreasing f (๐‘ฅ) = 2๐‘ฅ3 โ€“ 3๐‘ฅ2 โ€“ 36๐‘ฅ + 7 Step 1: Calculating fโ€™(๐‘ฅ) f (๐‘ฅ) = 2๐‘ฅ3 โ€“ 3๐‘ฅ2 โ€“ 36๐‘ฅ + 7 fโ€™ (๐‘ฅ) = 6๐‘ฅ2 โ€“ 6๐‘ฅ โ€“ 36 + 0 fโ€™ (๐‘ฅ) = 6 (๐‘ฅ2 โ€“ ๐‘ฅ โ€“ 6 ) fโ€™ (๐‘ฅ) = 6(๏ท๐‘ฅ๏ทฎ2๏ทฏ โ€“ 3๐‘ฅ + 2๐‘ฅ โ€“ 6) = 6(๐‘ฅ โ€“ 3) (๐‘ฅ + 2) Step 2: Putting fโ€™(๐‘ฅ) = 0 6(๐‘ฅ + 2) )(๐‘ฅ โ€“ 3) = 0 (๐‘ฅ + 2) (๐‘ฅ โ€“ 3) = ๏ท0๏ทฎ6๏ทฏ (๐‘ฅ + 2) (๐‘ฅ โ€“ 3) = 0 So, x = โ€“ 2 & x = 3 Step 3: Plotting point on real line โˆด Points โ€“2 , & 3 divide the real line into 3 disjoint intervals i.e. ( โ€“ ๐•”ด โˆ’2) ( โ€“2 , 3) & (3 , ๐•”ด Step 4: (a) f is strictly increasing in ( โ€“๐•”ด , โ€“2) & (3 , ๐•”ด) (b) f is strictly decreasing in ( โ€“2 , 3)

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