The maximum value of the expression sin θ cos^ 2 θ ( θ ∈[0, π]) is

As $sin^{2}\theta + cos^{2}\theta = 1,$

$⇒ sin^{2}\theta + \dfrac{cos^{2}\theta}{2} + \dfrac{cos^{2}\theta}{2} = 1$

Applying $AM \ge GM,$ we have

$\dfrac{sin^{2}\theta + \dfrac{cos^{2}\theta}{2} + \dfrac{cos^{2}\theta}{2}}{3} \ge {\left(\dfrac{sin^{2}\theta cos^{4}\theta}{4}\right)}^{\dfrac{1}{3}}$

$⇒ {\left(\dfrac{1}{3}\right)}^{3} \ge \dfrac{sin^{2}\theta cos^{4}\theta}{4}$

$⇒ \left(sin \theta ⋅ cos^{2}\theta\right)2 \le \dfrac{4}{27}$

$⇒ sin \theta ⋅ cos^{2}\theta \le \dfrac{2}{3\sqrt{3}}$