Answer
(a) $f^{'}(x)=(0.3) [ e^{-0.03x} -0.03 x e^{-0.03x}] $
(b) $x=\frac{1}{0.03}=\frac{100}{3}$
(c) $f(\frac{100}{3})\approx3.68$
Work Step by Step
$f(x)=0.3xe^{-0.03x}$
(a)
Taking derivative with respect to x
$f^{'}(x)=\frac{d( 0.3xe^{-0.03x})}{dx}$
$f^{'}(x)=(0.3)\frac{d( xe^{-0.03x})}{dx}$
$f^{'}(x)=(0.3) [ e^{-0.03x} (\frac{d x}{dx}) + x \frac{d( e^{-0.03x} )}{dx}] $
$f^{'}(x)=(0.3) [ e^{-0.03x} -0.03x e^{-0.03x}] $
(b)
Since
$f^{'}(x)=(0.3) [ e^{-0.03x} -0.03x e^{-0.03x}] $
Put $f^{'}(x)=0$
$0=(0.3) [ e^{-0.03x} -0.03x e^{-0.03x}] $
$(0.3) [ e^{-0.03x} -0.03x e^{-0.03x}] =0$
$ [ e^{-0.03x} -0.03x e^{-0.03x}] =0$
$ e^{-0.03x} =0.03x e^{-0.03x} $
Since $e^{-0.03x}$ is non zero, it may be cancelled
$ 1 =0.03x $
$ 0.03x=1 $
$x=\frac{1}{0.03}=\frac{100}{3}$
(c)
Since
$f(x)=0.3xe^{-0.03x}$
Put $x=\frac{100}{3}$
$f(\frac{100}{3})=0.3( \frac{100}{3})e^{-0.03 \frac{100}{3}}$
$f(\frac{100}{3})=0.3( \frac{100}{3})e^{(-0.03 ) ( \frac{100}{3})}= 10e^{-1}\approx3.68$
