Consider the function $f(x)=4 x+4 x^{-1}$. For this function there are four important intervals: $(-\infty, A]$, $[A, B),(B, C]$, and $[C, \infty)$ where $A$, and $C$ are the critical numbers and the function is not defined at $B$. Find $A$ $0^{s}$ and $B$ and $C$
Solution
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 \(f'(x)=4-\frac{4}{x^2}\).
Step 4 ：Setting the derivative equal to zero gives us the critical numbers. Solving \(f'(x)=0\) 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 \((-\infty, -1]\), \([-1, 0)\), \((0, 1]\), and \([1, \infty)\).
Step 7 ：Finally, we find that \(A = -1\), \(B = 0\), and \(C = 1\).
Step 8 ：\(\boxed{A = -1, B = 0, C = 1}\)
