Question

The best approximation of the minimum value attained by $e^{-x}sin\left(100x\right)$ for $x>0$ is

• A. -0.954

Answer 1

The correct option is A -0.954
$f\left(x\right)=e^{-x}sin\left(100x\right),x\ge 0$
$=\frac{sin100x}{e^x}$
$\because e^{-x}$ has always $+ve$ value so product will be minimum only when $sin\left(100x\right)$ will be minimum.
Hence, $f\left(x\right)$ will be minimum when $sin\left(100x\right)=-1$ or
$100x=\frac{3\pi }{2}\Rightarrow x=\frac{3\pi }{200}$
minimum $f\left(x\right)=\left(-1\right)e^{\frac{-3\pi }{200}}=-0.954$.