Given the function \( f(x) = 9x^3 - 4x \), you might be interested in various analyses like finding the derivative, critical points, or evaluating the function at specific points. Let's start with some basics:

### 1. **Evaluation of the function:**
If you want to evaluate the function at a specific value of \( x \), you simply substitute \( x \) with that value in the equation.

### 2. **Derivative of the function:**
The derivative \( f'(x) \) gives the rate of change of the function with respect to \( x \). For the given function:
\[
f'(x) = \frac{d}{dx} \left(9x^3 - 4x\right) = 27x^2 - 4
\]

### 3. **Critical Points:**
To find the critical points, set the derivative equal to zero and solve for \( x \):
\[
27x^2 - 4 = 0
\]
\[
27x^2 = 4
\]
\[
x^2 = \\[
x^2 = \frac{4}{27}
\]
\[
x = \pm \frac{2}{\sqrt{27}} = \pm \frac{2}{3\sqrt{3}} = \pm \frac{2\sqrt{3}}{9}
\]

### 4. **Second Derivative:**
To determine the concavity of the function at the critical points, we can take the second derivative:
\[
f''(x) = \frac{d}{dx} \left(27x^2 - 4\right) = 54x
\]

The sign of \( f''(x) \) will indicate whether the function is concave up (if \( f''(x) > 0 \)) or concave down (if \( f''(x) < 0 \)) at those points.

### 5. **Behavior of the Function:**
You might also be interested in the general behavior of the function as \( x \) approaches positive or negative infinity, which can help you understand the end behavior of the polynomial.

Would you like more details on any of these points or have any specific questions?

### Relative Questions:
1. How do you evaluate the function \( f(x) = 9x^3 - 4x \) at \( x = 2 \)?
2. What are the critical points of the function \( f(x) = 9x^3 - 4x \)?
3. How do you find the inflection points for the function \( f(x) \)?
4. Can you explain how the second derivative test works to classify critical points?
5. What is the behavior of the function \( f(x) = 9x^3 - 4x \) as \( x \to \infty \)?

### Tip:
When analyzing polynomial functions, always consider finding the first and second derivatives to determine critical points and concavity, as this helps in sketching the graph of the function.

Explore the function f(x) = 9x^3 - 4x with detailed analysis on derivatives, critical points, concavity, and end behavior. Learn how to find the derivative, critical points, and understand the behavior of the function as x approaches infinity. Suitable for advanced high school and undergraduate students.

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