1. Chapter 6 Class 12 Application of Derivatives
2. Serial order wise

Transcript

Ex 6.2,9 Prove that ð¦ = ï·4ï·sinï·®ðï·¯ï·®(2 +ï· cosï·®ð)ï·¯ï·¯ â Î¸ is an increasing function of Î¸ inï·0,ï·ðï·®2ï·¯ï·¯ . ð¦ = ï·4ï·sinï·®ðï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯ï·¯âð We need to prove that function is increasing function of Î¸ in ï·0 , ï·ðï·®2ï·¯ï·¯ i.e. we need to prove that ï·ðð¦ï·®ðð ï·¯ > 0 for Î¸ â ï·0 , ï·ðï·®2ï·¯ï·¯ We have ð¦ = ï·4ï·sinï·®ðï·¯ï·®2 + ï·cosï·®ðï·¯ï·¯ â Î¸ ï·ðð¦ï·®ðð ï·¯ = ï·ðï·®ððï·¯ï·ï·4ï·sinï·®ðï·¯ï·®2 + ï·cosï·®ðï·¯ï·¯ âðï·¯ ï·ðð¦ï·®ðð ï·¯ = ï·ðï·®ððï·¯ï·ï·4ï·sinï·®ðï·¯ï·®2 +ï·cosï·®ðï·¯ï·¯ ï·¯â ï·ððï·®ððï·¯ ï·ðð¦ï·®ðð ï·¯ = ï·ðï·®ððï·¯ï·ï·4ï·sinï·®ðï·¯ï·®2 +ï·cosï·®ðï·¯ï·¯ ï·¯â 1 ï·ðð¦ï·®ðð¥ï·¯ = ï·ï·ï·4ï·sinï·®ðï·¯ï·¯ï·®â²ï·¯ï·2 + ï·cosï·®ðï·¯ï·¯âï·ï·2 + ï·cosï·®ðï·¯ï·¯ï·®â²ï·¯ï·4ï·sinï·®ðï·¯ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ â1 ï·ðð¦ï·®ðð¥ï·¯ = ï·4ï·cosï·®ðï·¯ ï·2 + ï·cosï·®ðï·¯ï·¯âï·0â ï·sinï·®ðï·¯ï·¯ï·4ï·sinï·®ðï·¯ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ â1 ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð + 4 ï·ððð 2ð + ð ðð2ðï·¯ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ â1 ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð + 4ï·1ï·¯ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ â1 ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð + 4ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ â1 ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð + 4ï·¯ â ï·ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð + 4ï·¯ â ï·4 + ððð 2 ð + 4 ððð ðï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð + 4ï·¯ â 4 â ððð 2ð â 4 ððð ðï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ ï·ðð¦ï·®ðð¥ï·¯ = ï·8ï·cosï·®ð â 4ï·cosï·®ðï·¯ + 4 â 4 â ððð 2ðï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ ï·ðð¦ï·®ðð¥ï·¯ = ï·4 cos ð + 0 â ððð 2ðï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ ï·ðð¦ï·®ðð¥ï·¯ = ï·4ï·cosï·®ð â ððð 2 ðï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ For the function to be increasing ï·ðð¦ï·®ððï·¯ = ï·ï·cosï·®ðï·¯ï·4 â ï·cosï·®ðï·¯ï·¯ï·®ï·2 + ï·cosï·®ðï·¯ï·¯2ï·¯ > 0 Since denominator is a square, We need to show that cos Î¸ ï·4âï·cosï·®ðï·¯ï·¯ > 0 for Î¸ â ï·0 , ï·ðï·®2ï·¯ï·¯ Now, 0 â¤ cos Î¸ â¤ 1 Multiplying by â1 â1 â¤ âcos Î¸ â¤ 0 Adding 4 both sides â1 + 4 â¤ â cos Î¸ +4 < 0 + 4 3 â¤ 4 â cos2 Î¸ < 4 Thus, (4 â cos2 Î¸ ) is positive And cos Î¸ is also positive Hence cos Î¸ (4 â cos2 Î¸) is also positive â cos Î¸ (4 â cos2 Î¸) > 0 for Î¸ â ï·0 , ï·ðï·®2ï·¯ ï·¯ Hence ð¦ = ï·4ï·sinï·®ðï·¯ï·®2 + ððð ð ï·¯ â Î¸ is an increasing function for Î¸ â ï·ð , ï·ðï·®ðï·¯ï·¯

Serial order wise