Let f:(0,2 π) arrow R be defined as f(x)= sin x ⋅ e sin 2 x+∫ limits0x e sin 2 t( sin t- cos t) d t. Then the number of points of local minima of f(x) is equal to

Question

Let f:(0,2 π) arrow R be defined as f(x)= sin x ⋅ e sin 2 x+∫ limits0x e sin 2 t( sin t- cos t) d t. Then the number of points of local minima of f(x) is equal to (A) 2 (B) 3 (C) 6 (D) 8

Tardigrade Admin · Accepted Answer

We have, f(x)= sin x ⋅ e sin 2 x Ix ∫ limits0x e sin 2 t( sin t- cos t) d t ∴ f prime( x )= sin x (2 cos 2 x ) ⋅ e sin 2 x +( cos x ) e sin 2 x + e sin 2 x ( sin x - cos x ) = e sin 2 x ⋅ sin x (2 cos 2 x +1) =2 ⋅ e sin 2 x ⋅ sin x ⋅( cos 2 x+(1/2)) =2 ⋅ e sin 2 x sin x ⋅ 2( cos 2 x-(1/4)) =4 ⋅ e sin 2 x sin x ⋅( cos x+(1/2)) ⋅( cos x-(1/2))

Q. Let f:(0,2π)→R be defined as f(x)=sinx⋅esin2x+0∫x​esin2t(sint−cost)dt. Then the number of points of local minima of f(x) is equal to

2

3

6

8

We have, f(x)=sinx⋅esin2xIx0∫x​esin2t(sint−cost)dt ∴f′(x)=sinx(2cos2x)⋅esin2x+(cosx)esin2x+esin2x(sinx−cosx) =esin2x⋅sinx(2cos2x+1) =2⋅esin2x⋅sinx⋅(cos2x+21​) =2⋅esin2xsinx⋅2(cos2x−41​) =4⋅esin2xsinx⋅(cosx+21​)⋅(cosx−21​)

Q. Let $f : (0, 2 π) \to R$ be defined as $f (x) = sin x \cdot e^{s i n 2 x} + 0 \int x e^{s i n 2 t} (sin t - cos t) d t$ . Then the number of points of local minima of $f (x)$ is equal to