max-sin-x-cos-3-x-x-0-pi-2-

Question Number 174271 by mnjuly1970 last updated on 28/Jul/22

m a x (s i n (x) . {c o s}^{3} (x)) = ?

x \in [0, \frac{π}{2}]

Commented by blackmamba last updated on 28/Jul/22

f (x) = \sin x (1 - \sin^{2} x) \cos x

f (x) = \frac{1}{2} \sin 2 x (1 - (\frac{1 - \cos 2 x}{2}))

f (x) = \frac{1}{4} \sin 2 x (1 + \cos 2 x)

f (x) = \frac{1}{4} \sin 2 x + \frac{1}{8} \sin 4 x

f^{'} (x) = \frac{1}{2} \cos 2 x + \frac{1}{2} \cos 4 x = 0

\Rightarrow 2 \cos 3 x \cos x = 0

{\begin{cases} \cos 3 x = 0 \Rightarrow 3 x = \frac{π}{2}; x = \frac{π}{6} \\ \cos x = 0 \Rightarrow x = \frac{π}{2} \end{cases}

{\begin{cases} f o r x = \frac{π}{6} \Rightarrow f (\frac{π}{6}) = \frac{1}{4} \sin (\frac{π}{3}) [1 + \cos (\frac{π}{3})] \\ f o r x = \frac{π}{2} \Rightarrow f (\frac{π}{2}) = \frac{1}{4} \sin π [1 + \cos π] = 0 \end{cases}

m a x = f (\frac{π}{6}) = \frac{1}{4} \times \frac{1}{2} \sqrt{3} \times \frac{3}{2} = \frac{3 \sqrt{3}}{16}

Commented by mnjuly1970 last updated on 28/Jul/22

t h x a l o t S i r