The minimum value of 27 cos 2x 81 sin 2x is

Question

The minimum value of 27 cos 2x 81 sin 2x is (A)  (1/27) (B) -5 (C)  (1/5) (D)  (1/243)

Tardigrade Admin · Accepted Answer

Let f(x) =27 cos 2 x 81 sin 2 x=33 cos 2 x+4 sin 2 x =35((3/5) cos 2 x+(4/5) sin 2 x) Let (3/5)= sin φ ⇒ (4/5)= cos φ then f(x)=35( sin φ cos 2 x+ cos φ sin 2 x) =35( sin (φ+2 x)) For minimum value of given function, sin (φ+2 x) will be minimum, ie, sin (φ+2 x)=-1 ∴ f(x)=35(-1)=(1/243)

Q. The minimum value of $2 7^{c o s 2 x} 8 1^{s i n 2 x}$ is

$\frac{1}{27}$

$- 5$

$\frac{1}{5}$

$\frac{1}{243}$

Q. The minimum value of 27cos2x81sin2x is

271​

−5

51​

2431​

Let f(x)=27cos2x81sin2x=33cos2x+4sin2x =35(53​cos2x+54​sin2x) Let 53​=sinϕ ⇒54​=cosϕ then f(x)=35(sinϕcos2x+cosϕsin2x) =35(sin(ϕ+2x)) For minimum value of given function, sin(ϕ+2x) will be minimum, ie, sin(ϕ+2x)=−1 ∴f(x)=35(−1)=2431​

Q. The minimum value of $2 7^{c o s 2 x} 8 1^{s i n 2 x}$ is

$\frac{1}{27}$

$- 5$

$\frac{1}{5}$

$\frac{1}{243}$