Example 48 - Chapter 6 Class 12 Application of Derivatives
Last updated at Jan. 7, 2020 by Teachoo
Transcript
Example 48 Show that the function f given by f (x) = tanβ1(sin x + cos x), x > 0 is always an strictly increasing function in (0,π/4) f(π₯)=tan^(β1)β‘(sinβ‘π₯+cosβ‘π₯ ) Finding fβ(π) fβ(π₯) = (π(tan^(β1)β‘γ(sinβ‘π₯ +cosβ‘π₯ ))γ)/ππ₯ = 1/(1 + (sinβ‘γπ₯ + cosβ‘π₯ γ )^2 ) Γ π(sinβ‘γπ₯ + cosβ‘π₯ γ )/ππ₯ = 1/(1 + (γγπ¬π’π§γ^π πγβ‘γ+γππ¨π¬γ^πβ‘γπ γ+ 2 sinβ‘π₯ cosβ‘π₯ γ ) ) Γ (cosβ‘π₯βsinβ‘π₯ ) = 1/(1 + (π + 2 sinβ‘π₯ cosβ‘π₯ ) ) Γ (cosβ‘π₯βsinβ‘π₯ ) = 1/(2 + 2 sinβ‘π₯ cosβ‘π₯ ) Γ (cosβ‘π₯βsinβ‘π₯ ) ((tan^(β1)β‘π₯ )^β²= 1/(1 + π₯^2 )) (sin2 x + cos2 x = 1) = cosβ‘γπ₯ βγ sinγβ‘π₯ γ/(2 + sinβ‘2π₯ ) For increasing, fβ(x) > 0. β΄ Numerator and denominator both must be > 0 Checking sign for denominator Hence, denominator is always positive for 0 < x < π/4 (As sin2x = 2 sin x cos x) Checking sign for numerator cos π₯ β sin π₯ > 0. cos π₯ > sin π₯. cosβ‘π₯/cosβ‘π₯ > sinβ‘π₯/cosβ‘γπ₯.γ 1 > tan π₯ tanβ‘π₯<1 That is possible only if 0<π₯<π/4 Thus, fβ(π₯) = ((+))/((+) ) > 0 in x β (0 , π/4) Hence f is strictly increasing function is (π , π /π)
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Chapter 6 Class 12 Application of Derivatives
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Davneet Singh
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.