Let f(x)=3 sin 4 x+10 sin 3 x+6 sin 2 x-3, x ∈[-(π/6), (π/2)] . Then, f is :

Question

Let f(x)=3 sin 4 x+10 sin 3 x+6 sin 2 x-3, x ∈[-(π/6), (π/2)] . Then, f is : (A) increasing in (-(π/6), (π/2)) (B) decreasing in (0, (π/2)) (C) increasing in (-(π/6), 0) (D) decreasing in (-(π/6), 0)

Tardigrade Admin · Accepted Answer

f(x)=3 sin 4 x+10 sin 3 x+6 sin 2 x-3, x ∈[-(π/6), (π/2)] f prime(x)=12 sin 3 x cos x+30 sin 2 x cos x+12 sin x cos x =6 sin x cos x(2 sin 2 x+5 sin x+2) =6 sin x cos x(2 sin x+1)( sin +2) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/3f21dd5ff806c919b0b383fa1b573ad4-.png" /> decreasing in (-(π/6), 0)

Q. Let $f (x) = 3 sin^{4} x + 10 sin^{3} x + 6 sin^{2} x - 3, x \in [- \frac{π}{6}, \frac{π}{2}] .$ Then, $f$ is :

increasing in $(- \frac{π}{6}, \frac{π}{2})$

decreasing in $(0, \frac{π}{2})$

increasing in $(- \frac{π}{6}, 0)$

decreasing in $(- \frac{π}{6}, 0)$

$f (x) = 3 sin^{4} x + 10 sin^{3} x + 6 sin^{2} x - 3, x \in [- \frac{π}{6}, \frac{π}{2}]$
$f^{'} (x) = 12 sin^{3} x cos x + 30 sin^{2} x cos x + 12 sin x cos x$
$= 6 sin x cos x (2 sin^{2} x + 5 sin x + 2)$
$= 6 sin x cos x (2 sin x + 1) (sin + 2)$

decreasing in $(- \frac{π}{6}, 0)$

Q. Let f(x)=3sin4x+10sin3x+6sin2x−3,x∈[−6π​,2π​]. Then, f is :

increasing in (−6π​,2π​)

decreasing in (0,2π​)

increasing in (−6π​,0)

decreasing in (−6π​,0)