Check sibling questions

Example 48 - Show that f(x) = tan-1 (sin x + cos x) is always

Example 48 - Chapter 6 Class 12 Application of Derivatives - Part 2
Example 48 - Chapter 6 Class 12 Application of Derivatives - Part 3

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Example 34 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 ) = 𝐜𝐨𝐬⁡〖𝒙 −〖 𝐬𝐢𝐧〗⁡𝒙 〗/(𝟐 + 𝐬𝐢𝐧⁡𝟐𝒙 ) For increasing, f’(x) > 0. ∴ Numerator and denominator both must be > 0 Checking sign for denominator Since Hence, denominator is always positive for 0 < x < 𝝅/𝟒 Checking sign for numerator cos 𝑥 – sin 𝑥 > 0 cos 𝑥 > sin 𝑥 1 > sin⁡𝑥/cos⁡𝑥 1 > tan 𝑥 𝐭𝐚𝐧⁡𝒙<𝟏 This is possible only if 𝟎<𝒙<𝝅/𝟒 Thus, f’(𝒙) = ((+))/((+) ) > 0 in x ∈ (0 , 𝜋/4) Hence, f is strictly increasing function in (𝟎 , 𝝅/𝟒)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.