Question

The inequality.
$minf\left(x\right)>-7/18[-\pi ,\pi ]$ holds for the function $f\left(x\right)=\left(cosx{)}^2\right(sinx).$

• A. True
• B. False

Answer 1

The correct option is A True

$f\left(x\right)=(\cos x{)}^2\sin x$

$f^{\prime }\left(x\right)=2\cos x(-\sin x)\sin x+(\cos x{)}^2.\cos x$

$=2\cos x{\sin }^2\theta +{\cos }^{3}x$

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

$=\cos x(1-3{\sin }^{2}x)$

for roots $\underset{\sin x=1}{=\cos x=0}$ as $\underset{{\cos }^{2}x=\frac{2}{3}}{{\sin }^{2}x=\frac{1}{3}}$

for ${\sin }^{2}x=\frac{1}{3},{\cos }^{2}x=\frac{2}{3}$

$f\left(x\right)={\cos }^{2}xsinx$

$=\frac{2}{3}\left(\frac{-1}{\sqrt{3}}\right)=\frac{-2}{3\sqrt{3}}=-0.3853>-0.38888=\frac{-7}{18}$