Example: Let the function $f(x)=\sqrt{x^{2}+1}-a x$, where $a>0$. Find the range of values for $a$ such that the function $f(x)$ is monotonic on $[0,+\infty)$.
(1) For any $0 \leqslant x_{1}<x_{2}$, if $f\left(x_{1}\right)>f\left(x_{2}\right)$ when $a \geqslant 1$, then $f(x)$ is a monotonically decreasing function on the interval $[0,+\infty)$.
(2) When $0<a<1$, there exist two points $x_{1}=0, x_{2}=\frac{2 a}{1-a^{2}}$ in the interval $[0,+\infty)$, such that $f\left(x_{1}\right)=$ $f\left(x_{2}\right)=1$, hence $f(x)$ is not a monotonic function on $[0,+\infty)$.
In summary, $f(x)$ is a monotonic function on the interval $[0,+\infty)$ if and only if $a \geqslant 1$.