Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12

Last updated at Jan. 7, 2020 by Teachoo

Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12

Transcript

Ex 6.2,13 On which of the following intervals is the function f given by f (π₯) = π₯^100 + sinβ‘π₯ β1 strictly decreasing ? (A) (0, 1) (B) (π/2, π) (C) (0, π/2) (D) None of these f (x) = π₯100 + sin x β 1 fβ(x) = 100π₯99 + cos x (A) (0, 1) For 0 < x < 1 0 < x < 1.57 099 < x99 < (1.57)99 0 < x99 < 2.47 Γ 1099 0 < 100x99 < 247 Γ 1019 0 < x < 1 β΄ 0 < x < π/2 So x is in 1st quadrant β΄ cos x is positive (As π/2 = 3.14/2 = 1.57 > 1) So, 100 x99 + cos x is positive. β΄ fβ(x) > 0 f (x) is increasing on (0, 1) (B) (π /π, π ) π/2 < x < π 1.57 < x < 3.14 Since 100 x99 is much greater than β1 So 100 x99 + cos x is positive β΄ fβ (x) > 0 1.57 < x < 3.14 (1.57)99 < x99 < (3.14)99 2.47 Γ 1099 < x99 < 3.14 Γ 1099 247 Γ 1099 < 100 x99 < 314 Γ 1099 Since π/2 < x < π so x is in 2nd quadrant β΄ cos x is negative. Minimum value of cos x = β1 (πͺ)(π, π /π) 0<π₯< π/2 So 100 x99 + cos x is positive β΄ fβ (x) > 0 f(x) is increasing on (0,π/2) 0<π₯< π/2 0 < x < 1.57 099 < x99 < (1.57)99 0 < x99 < 2.47 Γ 1099 0 < cos x < π/2 So x is in 1st quadrant β΄ cos x is positive Thus, f(x) is strictly decreasing for none of the intervals. So, (D) is the correct answer

Chapter 6 Class 12 Application of Derivatives

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.