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, 9 Prove that π¦ = (4 sinβ‘π)/((2 +γ cosγβ‘γπ)γ ) β ΞΈ is an increasing function of ΞΈ in[0,π/2] . π¦ = (4 sinβ‘π)/((2 + cosβ‘π ) )βπ We need to prove that function is increasing function of ΞΈ in (0 , π/2) i.e. we need to prove that ππ¦/(ππ ) > 0 for ΞΈ β [0 , π/2] We have π¦ = (4 sinβ‘π)/(2 + cosβ‘π ) β ΞΈ ππ¦/(ππ ) = π/ππ ((4 sinβ‘π)/(2 + cosβ‘π ) βπ) ππ¦/(ππ ) = π/ππ ((4 sinβ‘π)/(2 +cosβ‘π ) )β ππ/ππ ππ¦/(ππ ) = π/ππ ((4 sinβ‘π)/(2 +cosβ‘π ) )β 1 ππ¦/ππ₯ = ((4 sinβ‘π )^β² (2 + cosβ‘π ) β (2 + cosβ‘π )^β² (4 sinβ‘π ))/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (4 cosβ‘π (2 + cosβ‘π ) β (0 β sinβ‘π )(4 sinβ‘π ))/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (8 cosβ‘γπ + 4 (πππ 2π + π ππ2π)γ)/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (8 cosβ‘γπ + 4(1)γ)/(2 + cosβ‘π )2 β1 Using quotient rule as (π’/π£)^β²= (π’^β² π£ β π£^β² π’)/π£2 Where u = 4 sinβ‘π & π£= 2+cosβ‘π ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ)/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ β (2 + cosβ‘π )2)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ β (4 + πππ 2 π + 4 πππ π))/(2 + cosβ‘π )2 ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ β 4 β πππ 2π β 4 πππ π)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (8 cosβ‘γπ β 4 cosβ‘π + 4 β 4 β πππ 2πγ)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (4 cos π + 0 β πππ 2π)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (4 cosβ‘γπ β πππ 2 πγ)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (cosβ‘π (4 β cosβ‘π ))/(2 + cosβ‘π )2 For the function to be increasing ππ¦/ππ = (cosβ‘π (4 β cosβ‘π ))/(2 + cosβ‘π )2 > 0 Since denominator is a square, We need to show that cos ΞΈ (4βcosβ‘π ) > 0 for ΞΈ β [0 , π/2] Now, 0 β€ cos ΞΈ β€ 1 Multiplying by β1 β1 β€ βcos ΞΈ β€ 0 Adding 4 both sides β1 + 4 β€ β cos ΞΈ + 4 < 0 + 4 3 β€ 4 β cos ΞΈ < 4 Thus, (4 β cos ΞΈ) is positive And cos ΞΈ is also positive Hence cos ΞΈ (4 β cos2 ΞΈ) is also positive β cos ΞΈ (4 β cos2 ΞΈ) > 0 for ΞΈ β [0 , π/2 ] Hence π¦ = (4 sinβ‘π)/(2 +cosβ‘π ) β ΞΈ is an increasing function for ΞΈ β [π , π /π]
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