Ex 6.2

Chapter 6 Class 12 Application of Derivatives
Serial order wise

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

### Transcript

Ex 6.2, 9 Prove that π¦ = (4 sinβ‘π)/((2 +γ cosγβ‘γπ)γ ) β ΞΈ is an increasing function of ΞΈ in[0,π/2] . π¦ = (4 sinβ‘π)/((2 + cosβ‘π ) )βπ We need to prove that function is increasing function of ΞΈ in (0 , π/2) i.e. we need to prove that ππ/(ππ½ ) > 0 for ΞΈ β [π , π/π] Differentiating w.r.t ΞΈ ππ¦/(ππ ) = π/ππ ((4 sinβ‘π)/(2 + cosβ‘π ) βπ) ππ¦/(ππ ) = π/ππ ((4 sinβ‘π)/(2 +cosβ‘π ) )β ππ/ππ ππ¦/(ππ ) = π/ππ ((4 sinβ‘π)/(2 +cosβ‘π ) )β 1 ππ¦/ππ₯ = ((π πππβ‘π½ )^β² (π + πππβ‘π½ ) β (π + πππβ‘π½ )^β² (π πππβ‘π½ ))/(π + πππβ‘π½ )π β1 ππ¦/ππ₯ = (4 cosβ‘π (2 + cosβ‘π ) β (0 β sinβ‘π )(4 sinβ‘π ))/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (8 cosβ‘γπ + 4 (πππππ½ + πππππ½)γ)/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (8 cosβ‘γπ + 4(π)γ)/(2 + cosβ‘π )2 β1 ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ β (2 + cosβ‘π )2)/(2 + cosβ‘π )2 Using quotient rule as (π’/π£)^β²= (π’^β² π£ β π£^β² π’)/π£2 Where u = 4 sinβ‘π & π£= 2+cosβ‘π ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ β (4 + πππ 2 π + 4 πππ π))/(2 + cosβ‘π )2 ππ¦/ππ₯ = (8 cosβ‘γπ + 4γ β 4 β πππ 2π β 4 πππ π)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (8 cosβ‘γπ β 4 cosβ‘π + 4 β 4 β πππ 2πγ)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (4 cos π + 0 β πππ 2π)/(2 + cosβ‘π )2 ππ¦/ππ₯ = (4 cosβ‘γπ β πππ 2 πγ)/(2 + cosβ‘π )2 ππ/ππ = (πππβ‘π½ (π β πππβ‘π½ ))/(π + πππβ‘π½ )π For the function to be increasing ππ¦/ππ > 0 (ππ¨π¬β‘π½ (π β πππβ‘π½ ))/(π + πππβ‘π½ )π > 0 Since denominator is a square, We need to show that cos ΞΈ (πβππ¨π¬β‘π½ ) > 0 for ΞΈ β [0 , π/2] Now, 0 β€ cos ΞΈ β€ 1 Multiplying by β1 β1 β€ βcos ΞΈ β€ 0 Adding 4 both sides β1 + 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]