Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12   1. Chapter 6 Class 12 Application of Derivatives
2. Serial order wise
3. Examples

Transcript

Example 48 Show that the function f given by f (x) = tan–1(sin x + cos x), x > 0 is always an strictly increasing function in (0,𝜋/4) f(𝑥)=tan^(−1)⁡(sin⁡𝑥+cos⁡𝑥 ) Finding f’(𝒙) f’(𝑥) = (𝑑(tan^(−1)⁡〖(sin⁡𝑥 +cos⁡𝑥 ))〗)/𝑑𝑥 = 1/(1 + (sin⁡〖𝑥 + cos⁡𝑥 〗 )^2 ) × 𝑑(sin⁡〖𝑥 + cos⁡𝑥 〗 )/𝑑𝑥 = 1/(1 + (〖〖𝐬𝐢𝐧〗^𝟐 𝒙〗⁡〖+〖𝐜𝐨𝐬〗^𝟐⁡〖𝒙 〗+ 2 sin⁡𝑥 cos⁡𝑥 〗 ) ) × (cos⁡𝑥−sin⁡𝑥 ) = 1/(1 + (𝟏 + 2 sin⁡𝑥 cos⁡𝑥 ) ) × (cos⁡𝑥−sin⁡𝑥 ) = 1/(2 + 2 sin⁡𝑥 cos⁡𝑥 ) × (cos⁡𝑥−sin⁡𝑥 ) ((tan^(−1)⁡𝑥 )^′= 1/(1 + 𝑥^2 )) (sin2 x + cos2 x = 1) = cos⁡〖𝑥 −〖 sin〗⁡𝑥 〗/(2 + sin⁡2𝑥 ) For increasing, f’(x) > 0. ∴ 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 𝑥 – sin 𝑥 > 0. cos 𝑥 > sin 𝑥. cos⁡𝑥/cos⁡𝑥 > sin⁡𝑥/cos⁡〖𝑥.〗 1 > tan 𝑥 tan⁡𝑥<1 That is possible only if 0<𝑥<𝜋/4 Thus, f’(𝑥) = ((+))/((+) ) > 0 in x ∈ (0 , 𝜋/4) Hence f is strictly increasing function is (𝟎 , 𝝅/𝟒)

Examples 