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
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Ex 6.2,12 Which of the following functions are strictly decreasing on (0 ,π/2)? (A) cos π₯ Let f(π₯) = cos π₯ Finding fβ(π₯) fβ(π₯) = β sin π₯ Hence, fβ(π₯) < 0 for π₯ β (0 , π/2) β f(π₯) is strictly decreasing on (π , π /π) Note :- sin π₯ > 0 for π₯ β (0 , π/2) So, β sin π₯ < for π₯ β (0 , π/2) fβ (π₯)<0 πππ π₯ β π/2 Ex 6.2,12 Which of the following functions are strictly decreasing on (0,π/2) ? (B) cos 2π₯ Let f(π₯) = cos 2π₯ Finding fβ(π) fβ(π₯) = (cosβ‘2π₯ )β² fβ(π₯) = β 2 sin 2π₯ Putting fβ(π) = 0 fβ(π₯) = 0 β 2 sin 2π₯ = 0 sin 2π₯ = 0 As π₯ β (0 , π/2) 0 < π₯ < π/2 0 Γ 2 < 2π₯ < π/2 Γ 2 0 < 2π₯ < Ο We know that sin ΞΈ > 0 for ΞΈ β (0 , π) sin 2π₯ > 0 for 2π₯ β (0 , π) βsin 2π₯ < 0 for 2π₯ β (0 , π) β 2 sin 2π₯ < 0 for π₯ β (0 , π/2) β fβ(π₯) < 0 for π₯ β (0 , π/2) β f (π₯) is strictly decreasing for π β (π , π /π) Ex 6.2,12 Which of the following functions are strictly decreasing on (0,π/2) ? (C) cos 3π₯ Let f(π₯) = cos 3π₯ as π₯ β (0 ,π/2) Finding fβ(π) fβ(π₯) = (cosβ‘3π₯ )^β² fβ(π₯) = β3 sin 3π₯ Putting fβ(π) = 0 β3 sin 3π₯ = 0 sin 3π₯ = 0 As sin ΞΈ = 0 if ΞΈ = 0 , Ο , 2Ο , 3Ο β΄ 3π₯ = Ο , 2Ο π₯ = π/3 , 2π/3 As π₯ β (0 , π/2) So, π₯ = π/3 Since π₯ β (0 , π/2), so we start number line from (0 , π/2) The point x = π/3 divide the interval (0 , π/2) into 2 disjoint interval. i.e. (0 , π/3) & (π/3 , π/2) Checking sign of fβ(π₯) Case 1: For π β (π , π /π) Now, 0 < π₯ < π/3 3 Γ 0 < 3π₯ < 3π/3 0 < 3π₯ < Ο So, when π₯ β (0 , π/3) , 3π₯ β (0 , π) We know that sinΞΈ > 0 for ΞΈ β (0 , π) sin 3π₯ > 0 for 3π₯ β (0 , π) sin 3π₯ > 0 for π₯ β (0 , π/3) β sin 3π₯ < 0 for π₯ β (0 ,π/3) β fβ(π₯) < 0 for π₯ β (0 , π/3) β f(π₯) is strictly decreasing for π₯ β (0 ,π/3) Case 2: For π β (π /π, π /π) Since π/3 < π₯ < π/2 3 Γ π/3 < 3π₯ < π/2 Γ 3 Ο < 3π₯ < 3π/2 We know that sin ΞΈ < 0 in 3rd quadrant sin ΞΈ < 0 for ΞΈ β (π, 3π/2) sin 3x < 0 for 3x β (π, 3π/2) sin 3x < 0 for x β (π/3, 3π/(2 Γ 3)) sin 3π₯ < 0 for π₯ β (π/3, π/2) βsin 3π₯ > 0 for π₯ β (π/3, π/2) β 3sin 3π₯ > 0 for π₯ β (π/3, π/2) fβ(π₯) > 0 for π₯ β (π/3, π/2) Hence f(π₯) is strictly increasing for π₯ β (π/3, π/2) Thus f(π₯) is neither decreasing nor increasing on π β (π , π /π) Ex 6.2, 12 Which of the following functions are strictly decreasing on (0,π/2) ? (D) tan π₯ Let f(π₯) = tan π₯ So, fβ(π₯) = sec^2 π₯ As square of any number is always positive So, fβ(π₯) > 0 for any value of π₯ Hence fβ(π₯) > 0 for x β (0 , π/2) β΄ f(π₯) is strictly increasing function Hence, (π¨) & (π©) is strictly decreasing on (0 , π/2)
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