#### Answer

$f$ has 14 critical numbers.

#### Work Step by Step

$f'(x) = \frac{100~cos^2~x}{10+x^2}-1$
To find the critical numbers of $f$, 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 $f'(x) = 0$:
$f'(x) = \frac{100~cos^2~x}{10+x^2}-1 = 0$
$\frac{100~cos^2~x}{10+x^2}= 1$
$100~cos^2~x = 10+x^2$
The left side of the equation has a maximum value of $100$
The right side of the equation is greater than $100$ when $x \lt -10$ or $x \gt 10$
Therefore, to find the points where $f'(x) = 0$, we can graph the function $f'(x)$ in the interval $[-10, 10]$ to find the points where $f'(x)$ crosses the x-axis.
On the sketch of the graph of $f'(x)$, we can see that the function $f'(x)$ has $14$ x-intercepts.
Therefore, $f$ has 14 critical numbers.