Consider the function $$ f(x)=9 x+2 x^{-1} $$. 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 $$ A $$ and $$ B \square $$ and $$ C $$

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To find the critical numbers $$ A $$ and $$ C $$ and the point $$ B $$ where the function $$ f(x) = 9x + 2x^{-1} $$ is not defined, we will follow these steps:

1. **Find the derivative of the function** to locate the critical points.
2. **Set the derivative equal to zero** to find $$ A $$ and $$ C $$.
3. **Identify where the function is not defined** to find $$ B $$.

Let's start by calculating the derivative of the function $$ f(x) $$.

The function is given as:
$$
f(x) = 9x + 2x^{-1}
$$

Now, let's find the derivative $$ f'(x) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$9+2\left(-1\right)x^{-2}$$
- step1: Multiply the first two terms:
  $$9-2x^{-2}$$
- step2: Simplify:
  $$9-\frac{2}{x^{2}}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{9x^{2}}{x^{2}}-\frac{2}{x^{2}}$$
- step4: Transform the expression:
  $$\frac{9x^{2}-2}{x^{2}}$$
The derivative of the function is given by:

$$
f'(x) = \frac{9x^{2} - 2}{x^{2}}
$$

Next, we will set the derivative equal to zero to find the critical points $$ A $$ and $$ C $$:

$$
9x^{2} - 2 = 0
$$

Now, let's solve for $$ x $$.
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
  $$9x^{2}-2=0$$
- step1: Add or subtract both sides:
  $$9x^{2}=2$$
- step2: Divide both sides:
  $$\frac{9x^{2}}{9}=\frac{2}{9}$$
- step3: Divide the numbers:
  $$x^{2}=\frac{2}{9}$$
- step4: Simplify the expression:
  $$x=\pm \sqrt{\frac{2}{9}}$$
- step5: Simplify the expression:
  $$x=\pm \frac{\sqrt{2}}{3}$$
- step6: Separate into possible cases:
  $$\begin{align}&x=\frac{\sqrt{2}}{3}\&x=-\frac{\sqrt{2}}{3}\end{align}$$
- step7: Rewrite:
  $$x_{1}=-\frac{\sqrt{2}}{3},x_{2}=\frac{\sqrt{2}}{3}$$
The critical points are:

$$
A = -\frac{\sqrt{2}}{3} \quad \text{and} \quad C = \frac{\sqrt{2}}{3}
$$

Next, we need to find the point $$ B $$ where the function is not defined. The function $$ f(x) = 9x + 2x^{-1} $$ is not defined when the denominator of the term $$ 2x^{-1} $$ is zero, which occurs when $$ x = 0 $$.

Thus, we have:

$$
B = 0
$$

In summary, the values are:
- $$ A = -\frac{\sqrt{2}}{3} $$
- $$ B = 0 $$
- $$ C = \frac{\sqrt{2}}{3} $$
$$ A = -\frac{\sqrt{2}}{3} $$, $$ B = 0 $$, and $$ C = \frac{\sqrt{2}}{3} $$.

Consider the function . For this
function there are four important intervals: ,
, and where , and are the
critical numbers and the function is not defined at .
Find
and
and

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