Ex 6.2, 6 - Find intervals in which functions strictly increasing

Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 2
Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 3 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 4 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 5 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 6 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 7 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 8 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 9 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 10 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 11 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 12 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 13 Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives - Part 14

  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Ex 6.2, 6 Find the intervals in which the following functions are strictly increasing or decreasing: (a) ๐‘ฅ2 + 2๐‘ฅ โ€“ 5 f(๐‘ฅ) = ๐‘ฅ2 + 2๐‘ฅ โ€“ 5 Calculating fโ€™ (๐’™) fโ€™(๐‘ฅ) = 2๐‘ฅ + 2 fโ€™(๐‘ฅ) = 2 (๐‘ฅ + 1) Putting fโ€™ (๐’™) = 0 2 (๐‘ฅ + 1) = 0 (๐‘ฅ + 1) = 0 ๐’™ = โ€“1 Plotting point on real line Hence, f is strictly decreasing for (โˆ’โˆž ," โ€“1" ) f is strictly increasing for ( "โ€“1" ,โˆž) Ex 6.2, 6 Find the intervals in which the following functions are strictly increasing or decreasing: (b) 10 โ€“ 6๐‘ฅ โ€“ 2๐‘ฅ2 Calculating fโ€™(๐’™) f(๐‘ฅ) = 10 โ€“ 6๐‘ฅ โ€“ 2๐‘ฅ2 fโ€™(๐‘ฅ) = โ€“6 โ€“ 4๐‘ฅ fโ€™(๐‘ฅ) = โˆ’2(3 + 2๐‘ฅ) Putting fโ€™(๐’™) = 0 โˆ’2(3 + 2๐‘ฅ) = 0 3 + 2๐‘ฅ = 0 2๐‘ฅ = โˆ’3 ๐’™ = (โˆ’๐Ÿ‘)/๐Ÿ Plotting points on number line Hence, f is strictly decreasing for ๐’™ > (โˆ’๐Ÿ‘)/๐Ÿ f is strictly increasing for x < (โˆ’๐Ÿ‘)/๐Ÿ Ex 6.2, 6 Find the intervals in which the following functions are strictly increasing or decreasing: (c) โ€“2๐‘ฅ3 โ€“ 9๐‘ฅ2 โ€“ 12๐‘ฅ + 1 f(๐‘ฅ) = โ€“2๐‘ฅ3 โ€“ 9๐‘ฅ2 โ€“ 12๐‘ฅ + 1 Calculating fโ€™(๐’™) fโ€™(๐‘ฅ) = โ€“6๐‘ฅ2 โ€“18๐‘ฅ โ€“ 12 + 0 fโ€™(๐‘ฅ) = โ€“6(๐‘ฅ2+3๐‘ฅ+2) fโ€™(๐‘ฅ) = โ€“6(๐‘ฅ2+2๐‘ฅ+๐‘ฅ+2) fโ€™(๐‘ฅ) = โ€“6(๐‘ฅ(๐‘ฅ+2)+1(๐‘ฅ+2)) fโ€™(๐’™) = โ€“6(๐’™+๐Ÿ) (๐’™+๐Ÿ) Putting fโ€™(๐’™) = 0 โ€“ 6(๐‘ฅ+1) (๐‘ฅ+2) = 0 (๐‘ฅ+1) (๐‘ฅ+2) = 0 So, ๐‘ฅ = โ€“1 , โ€“2 Plotting points on number line Hence, f is strictly increasing for โ€“2 < ๐’™ < โ€“1 & strictly decreasing for ๐’™ < โ€“2 & ๐’™ > โ€“1 Hence, f is strictly increasing for โ€“2 < ๐’™ < โ€“1 & strictly decreasing for ๐’™ < โ€“2 & ๐’™ > โ€“1 Ex 6.2, 6 Find the intervals in which the following functions are strictly increasing or decreasing: (d) 6 โ€“ 9๐‘ฅ โ€“ ๐‘ฅ2 ๐‘“(๐‘ฅ) = 6 โ€“ 9๐‘ฅ โ€“ ๐‘ฅ2 Calculating fโ€™(๐’™) fโ€™(๐‘ฅ) = โ€“9 โ€“ 2๐‘ฅ fโ€™(๐‘ฅ) = โ€“(2๐‘ฅ+9) Putting fโ€™(๐’™) = 0 โ€“(2๐‘ฅ+9) = 0 2๐‘ฅ + 9 = 0 2๐‘ฅ = โ€“ 9 ๐’™ = (โˆ’๐Ÿ—)/๐Ÿ Plotting point ๐’™ = (โˆ’๐Ÿ—)/๐Ÿ on real line Hence f is strictly Increasing for ๐’™ < (โˆ’๐Ÿ—)/๐Ÿ & strictly Decreasing for ๐’™ > (โˆ’๐Ÿ—)/๐Ÿ Ex 6.2, 6 Find the intervals in which the following functions are strictly increasing or decreasing: (e) (๐‘ฅ + 1)^3 (๐‘ฅ โ€“ 3)^3 f(๐‘ฅ) = (๐‘ฅ+1)3 (๐‘ฅโˆ’3)3 Calculating fโ€™(๐’™) f(๐‘ฅ) = (๐‘ฅ+1)3 (๐‘ฅโˆ’3)3 fโ€™(๐‘ฅ)= ใ€–[(๐‘ฅ+1)^3]ใ€—^โ€ฒ (๐‘ฅโˆ’3)^3 +[(๐‘ฅโˆ’3)3]^โ€ฒ (๐‘ฅ+1)^3 fโ€™(๐‘ฅ)=3(๐‘ฅ+1)2(๐‘ฅโˆ’3)3 + 3(๐‘ฅโˆ’3)2(๐‘ฅ+1)3 fโ€™(๐‘ฅ)=3(๐‘ฅ+1)2(๐‘ฅโˆ’3)2 ((๐‘ฅโˆ’3)+ (๐‘ฅ+1)) fโ€™(๐‘ฅ)=3(๐‘ฅ+1)2(๐‘ฅโˆ’3)2 (2๐‘ฅโˆ’2) fโ€™(๐’™)= 6(๐’™+๐Ÿ)๐Ÿ (๐’™โˆ’๐Ÿ‘)๐Ÿ (๐’™โˆ’๐Ÿ) Putting fโ€™(๐’™)=๐ŸŽ 6(๐‘ฅ+1)2 (๐‘ฅโˆ’3)2 (๐‘ฅโˆ’1) = 0 Hence, ๐‘ฅ = โ€“1 , 3 & 1 Plotting values of x on real line. Note that: fโ€™(๐‘ฅ) = 6 (๐’™+๐Ÿ)^๐Ÿ (๐’™โˆ’๐Ÿ‘)^๐Ÿ (๐‘ฅโˆ’1) Hence, f is strictly increasing for 1 < ๐‘ฅ < 3 & ๐‘ฅ > 3 i.e. (1, 3) and (3, โˆž) f is strictly decreasing for ๐‘ฅ < โ€“1 & โˆ’1<๐‘ฅ< 1 i.e. (โ€“โˆž, โ€“1) and (โ€“ 1, 1)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.