max 0 ≤ x ≤ π x - 2 sin x cos x + 1 3 sin 3 x =

Step-by-step solution

Let f x = x - 2 sin x cos x + 1 3 sin 3 x ⇒ f ' x = 1 - 2   cos 2 x + cos 3 x ⇒ f " x = 4 sin 2 x - 3 sin 3 x For maxima/minima f ' x = 0 ⇒ 1 - 2 2 cos 2 x - 1 + 4 cos 3 x - 3 cos x = 0 ⇒ ( 2   cos x + 3 ) ( 2   cos x - 3 ) ( cos x - 1 ) = 0 cos x = - 3 2 , 3 2 , 1 x = 5 π 6 , π 6 , 0 f " 5 π 6 = - 2 3 - 3 < 0 f " π 6 = 2 3 - 3 > 0 f " ( 0 ) = 0 So x = 5 π 6 is local maxima point Maximum value of f x = f 5 π 6 = 5 &