Find the inflection point(s) of the function $f(x)=5 x * e^{-4 x}$.

Select one:
$f(x)$ has an inflection point at $x=0.5$
$f(x)$ has an inflection point at $x=1$
$f(x)$ has an inflection point at $x=5$ and $x=2$
$f(x)$ has an inflection point at $x=-0.5$ and $x=0.5$
Clear my choice

Step 1 ：Given the function \(f(x)=5 x * e^{-4 x}\), we need to find the inflection point(s).

Step 2 ：To do this, we first find the first and second derivatives of the function.

Step 3 ：The first derivative of the function is \(f'(x) = -20*x*e^{-4*x} + 5*e^{-4*x}\).

Step 4 ：The second derivative of the function is \(f''(x) = 80*x*e^{-4*x} - 40*e^{-4*x}\).

Step 5 ：We then set the second derivative equal to zero and solve for x to find the x-values of the inflection points.

Step 6 ：Solving \(80*x*e^{-4*x} - 40*e^{-4*x} = 0\) gives us \(x = 0.5\).

Step 7 ：Thus, the function \(f(x)=5 x * e^{-4 x}\) has an inflection point at \(x=0.5\).

Step 8 ：\(\boxed{f(x) \text{ has an inflection point at } x=0.5}\)