Consider the function $f(x)=4x+4x^{-1}$. For this function there are four important intervals: $(-\mathrm{\infty },A]$, $[A,B),(B,C]$, and $[C,\mathrm{\infty })$ where $A$, and $C$ are the critical numbers and the function is not defined at $B$. Find $A$ $0^s$ and $B$ and $C$

Step 1 ：Given the function $f(x)=4x+\frac{4}{x}$, we need to find the critical numbers and the value of x for which the function is undefined.

Step 2 ：The critical numbers of a function are the values of x where the derivative of the function is either 0 or undefined. So, we first need to find the derivative of the function.

Step 3 ：The derivative of the function $f(x)$ is $f^{\prime }(x)=4-\frac{4}{x^2}$.

Step 4 ：Setting the derivative equal to zero gives us the critical numbers. Solving $f^{\prime }(x)=0$ gives us $x=-1$ and $x=1$.

Step 5 ：The function is undefined when the denominator is zero. So, we also need to find the value of x for which the function is undefined. Solving $x=0$ gives us the undefined number.

Step 6 ：Therefore, the critical numbers are -1 and 1, and the function is undefined at 0. So, the intervals are $(-\mathrm{\infty },-1]$, $[-1,0)$, $(0,1]$, and $[1,\mathrm{\infty })$.

Step 7 ：Finally, we find that $A=-1$, $B=0$, and $C=1$.

Step 8 ：$\boxed{{\displaystyle A=-1,B=0,C=1}}$