Check sibling questions

The function f (x) = 4 sin 3 x – 6 sin 2 x + 12 sin x + 100 is strictly

(A)Increasing in (Ο€, 3Ο€/2)Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  (B) decreasing in (Ο€/2,Ο€)

(C) decreasing in ((-Ο€)/2 ", "Β  Ο€/2)Β  `Β  Β  Β  Β  Β  (D) decreasingΒ  in ("0, "Β  Ο€/2)

Slide75.JPG

Slide76.JPG
Slide77.JPG Slide78.JPG Slide79.JPG

Get live Maths 1-on-1 Classs - Class 6 to 12


Transcript

Question 20 The function f (x) = 4 sin3 x – 6 sin2 x + 12 sin x + 100 is strictly Increasing in (πœ‹, 3πœ‹/2) (B) decreasing in (πœ‹/2,πœ‹) (C) decreasing in ((βˆ’πœ‹)/2 ", " πœ‹/2) ` (D) decreasing in ("0, " πœ‹/2) f (π‘₯) = 4 sin^3⁑〖π‘₯βˆ’6 sin^2⁑〖π‘₯+12 sin⁑〖π‘₯+100γ€— γ€— γ€— Differentiating w.r.t 𝒙 f’ (𝒙) = 4 ×⁑〖3 〗⁑〖〖𝑠𝑖𝑛〗^2 π‘₯γ€— Γ— (sin⁑〖π‘₯)β€²γ€—βˆ’6 Γ— 2 sin⁑π‘₯Γ— (sin⁑π‘₯ )β€²+12 cos⁑π‘₯+0 = 12 sin^2 π‘₯ cos π‘₯ βˆ’ 12 sin π‘₯ cos π‘₯ + 12 cos π‘₯ = 12 cos π‘₯ (γ€–sin^2 π‘₯γ€—β‘γ€–βˆ’sin⁑〖π‘₯+1γ€— γ€— ) = 12 cos 𝒙 (γ€–γ€–π’”π’Šπ’γ€—^𝟐 𝒙〗⁑〖+ (πŸβˆ’π¬π’π§β‘π’™ )γ€— ) Now, we need to check sign of f’ (𝒙) Checking sign of γ€–π’”π’Šπ’γ€—^πŸβ‘γ€–π’™+(γ€–πŸβˆ’π’”π’Šπ’γ€—β‘π’™ ) γ€— Since, γ€–π’”π’Šπ’γ€—^𝟐 𝒙 β‰₯ 0 And βˆ’1 ≀ sin 𝒙 ≀ 1 1 β‰₯ βˆ’ sin π‘₯ β‰₯ βˆ’1 1 + 1 β‰₯ 1 βˆ’ sin π‘₯ β‰₯ 1 βˆ’ 1 2 β‰₯ 1 βˆ’ sin π‘₯ β‰₯ 0 0 ≀ 1 βˆ’ sin π‘₯ ≀ 2 So, (1 βˆ’ sin 𝒙) β‰₯𝟎 Therefore, γ€–π’”π’Šπ’γ€—^πŸβ‘γ€–π’™+(γ€–πŸβˆ’π’”π’Šπ’γ€—β‘π’™ ) γ€—β‰₯ 0 Either cos 𝒙β‰₯ 0 π‘₯ πœ–(0, πœ‹/2)βˆͺ(3πœ‹/2, 2πœ‹) i.e., 1st and 4th quadrant, which can be written as 𝒙 ∈ ((βˆ’π…)/𝟐,𝝅/𝟐) Or cos 𝒙 ≀ 0 π‘₯ πœ–(πœ‹/2,πœ‹)βˆͺ(πœ‹,3πœ‹/2) i.e., 2nd and 3rd quadrant, which can be written as 𝒙 ∈(𝝅/𝟐,πŸ‘π…/𝟐) Thus, f is increasing in the interval ((βˆ’π…)/𝟐,𝝅/𝟐) f is decreasing in the interval (𝝅/𝟐,πŸ‘π…/𝟐) Our options are (A) Increasing in (πœ‹, 3πœ‹/2) (B) decreasing in (πœ‹/2,πœ‹) (C) decreasing in ((βˆ’πœ‹)/2 ", " πœ‹/2) ` (D) decreasing in ("0, " πœ‹/2) As (πœ‹/2,πœ‹) lies in the interval (πœ‹/2,3πœ‹/2) So, f is decreasing in this interval (𝝅/𝟐,𝝅) So, the correct answer is (B)

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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.