# Ex 6.2,13 - Chapter 6 Class 12 Application of Derivatives

Last updated at April 14, 2021 by Teachoo

Last updated at April 14, 2021 by Teachoo

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 Letβs check sign of fβ(x) in different intervals (A) (0, 1) For 0 < x < 1 Checking sign of 100x99 and cos x Since 0 < x < 1 099 < x99 < (1)99 0 < x99 < 1 0 < 100x99 < 100 β΄ 100x99 is positive Since 0 < x < 1 As π/2 = 3.14/2 = 1.57 > 1 Therefore, 0 < x < π/2 So, x is in 1st quadrant β΄ cos x is positive 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 Since 1.57 < x < 3.14 (1.57)99 < x99 < (3.14)99 (1.57)99 Γ 100 < 100x99 < (3.14)99 Γ 100 Since π/2 < x < π So x is in 2nd quadrant β΄ cos x is negative. Minimum value of cos x = β1 (C) (π,π /π) 0 < x < π/2 0 < x < 1.57 So, 100 x99 + cos x is positive. β΄ fβ(x) > 0 f(x) is increasing on (0, 1) Since 0 < x < 1.57 099 < x99 < (1.57)99 0 Γ 100 < 100x99 < (1.57)99 Γ 100 0 < 100x99 < (1.57)99 Γ 100 Since 0 < 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 10 years. He provides courses for Maths and Science at Teachoo.