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Local Maxima

fx=sinx. cos3...

Question

$f (x) = sin x . {cos}^{3} x$ has a maximum at x=

\frac{π}{4}

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\frac{π}{3}

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\frac{π}{6}

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\frac{π}{2}

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Solution

The correct option is C $\frac{π}{6}$
$f^{'} (x) = {cos}^{4} x - 3 {sin}^{2} x {cos}^{2} x$
$= {cos}^{2} x ({cos}^{2} x - 3 {sin}^{2} x)$
$f^{'} (x) = 0$ at $tan = \frac{1}{\sqrt{3}}$
$x = \frac{π}{6}$
$f^{'} (x) > 0$ If $α x < \frac{π}{6}$
$f^{'} (x) < 0$ If $\frac{π}{6} < x < \frac{π}{2}$
So $f (x)$ is max at $x = \frac{π}{6}$

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