The minimum value of the function f(x)=(3 sin x-4 cos x-10)(3 sin x+4 cos x -10), is

Question

The minimum value of the function f(x)=(3 sin x-4 cos x-10)(3 sin x+4 cos x -10), is (A) 49 (B) (195-60 √2/2) (C) 84 (D) 45

Tardigrade Admin · Accepted Answer

f ( x )=9 sin 2 x -16 cos 2 x -10(3 sin x -4 cos x )- 10(3 sin x +4 cos x )+100 =25 sin 2 x -60 sin x +84 =(5 sin x -6)2+48 ∴ f ( x ) min occurs when sin x =1 minimum value =49

Q. The minimum value of the function $f (x) = (3 sin x - 4 cos x - 10) (3 sin x + 4 cos x$ $- 10)$ , is

49

$\frac{195 - 60 2}{2}$

84

45

$f (x) = 9 sin^{2} x - 16 cos^{2} x - 10 (3 sin x - 4 cos x) -$
$10 (3 sin x + 4 cos x) + 100$
$= 25 sin^{2} x - 60 sin x + 84$
$= (5 sin x - 6)^{2} + 48$
$∴ f (x)_{m i n}$ occurs when $sin x = 1$
minimum value $= 49$

Q. The minimum value of the function f(x)=(3sinx−4cosx−10)(3sinx+4cosx −10), is

49

2195−602​​

84

45

f(x)=9sin2x−16cos2x−10(3sinx−4cosx)− 10(3sinx+4cosx)+100 =25sin2x−60sinx+84 =(5sinx−6)2+48 ∴f(x)min​ occurs when sinx=1 minimum value =49

Q. The minimum value of the function $f (x) = (3 sin x - 4 cos x - 10) (3 sin x + 4 cos x$ $- 10)$ , is

$\frac{195 - 60 2}{2}$

$f (x) = 9 sin^{2} x - 16 cos^{2} x - 10 (3 sin x - 4 cos x) -$
$10 (3 sin x + 4 cos x) + 100$
$= 25 sin^{2} x - 60 sin x + 84$
$= (5 sin x - 6)^{2} + 48$
$∴ f (x)_{m i n}$ occurs when $sin x = 1$
minimum value $= 49$