Answer
$f$ has 10 critical numbers.
Work Step by Step
$f'(x) = 5~e^{-1\vert x \vert}~sin~x - 1$
To find the critical numbers, we need to find the values of $x$ where $f'(x) = 0$ or $f'(x)$ is undefined.
We can see that $f'(x)$ is defined for all values of $x$
We can find the values of $x$ where $f'(x) = 0$:
$f'(x) = 5~e^{-0.1\vert x \vert}~sin~x - 1 = 0$
$5~e^{-1\vert x \vert}~sin~x = 1$
$5~sin~x = e^{0.1\vert x \vert}$
The left side of the equation has a maximum value of $5$
The right side of the equation is greater than $5$ when $x \lt -17$ or $x \gt 17$
Therefore, to find the points where $f'(x) = 0$, we can graph the function $f'(x)$ in the interval $[-17, 17]$ to find the points where $f'(x)$ crosses the x-axis.
On the sketch of the graph, we can see that the function $f'(x)$ has 10 x-intercepts.
Therefore, $f$ has 10 critical numbers.