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Q.
The maximum value of the expression $sin \theta cos^{2}\theta \, \left(\forall \theta \in \left[0 , \pi \right]\right)$ is
NTA AbhyasNTA Abhyas 2020
Solution:
As $sin^{2}\theta +cos^{2}\theta =1,$
$\Rightarrow sin^{2}\theta +\frac{c o s^{2} \theta }{2}+\frac{c o s^{2} \theta }{2}=1$
Applying $AM\geq GM,$ we have
$\frac{s i n^{2} \theta + \frac{c o s^{2} \theta }{2} + \frac{c o s^{2} \theta }{2}}{3}\geq \left(\frac{s i n^{2} \theta c o s^{4} \theta }{4}\right)^{\frac{1}{3}}$
$\Rightarrow \left(\frac{1}{3}\right)^{3}\geq \frac{s i n^{2} \theta c o s^{4} \theta }{4}$
$\Rightarrow \left(sin \theta \cdot c o s^{2} \theta \right)^{2}\leq \frac{4}{27}$
$\Rightarrow sin \theta \cdot cos^{2}\theta \leq \frac{2}{3 \sqrt{3}}$