To analyze the function \( f(x) = 5x^3 - 8x^2 + 4x \), we can compute its key characteristics such as its derivative, critical points, and behavior. Here's a breakdown:

### 1. **First Derivative (f'(x))**: 
To find the critical points and intervals where the function increases or decreases, we calculate the derivative of \( f(x) \):

\[
f'(x) = \frac{d}{dx} (5x^3 - 8x^2 + 4x)
\]

\[
f'(x) = 15x^2 - 16x + 4
\]

### 2. **Critical Points**: 
Set \( f'(x) = 0 \) to find the critical points:

\[
15x^2 - 16x + 4 = 0
\]

Using the quadratic formula:

\[
x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(15)(4)}}{2(15)}
\]

\[
x = \frac{16 \pm \sqrt{256 - 240}}{30}
\]

\[
x = \frac{16 \pm \sqrt{16}}{30}
\]

\[
x = \frac{16 \pm 4}{30}
\]

The solutions are:

\[
x = \frac{16 + 4}{30} = \frac{20}{30} = \frac{2}{3}, \quad x = \frac{16 - 4}{30} = \frac{12}{30} = \frac{2}{5}
\]

So, the critical points are \( x = \frac{2}{3} \) and \( x = \frac{2}{5} \).

### 3. **Second Derivative (f''(x))**:
To determine concavity and points of inflection, we compute the second derivative:

\[
f''(x) = \frac{d}{dx} (15x^2 - 16x + 4) = 30x - 16
\]

Setting \( f''(x) = 0 \) to find inflection points:

\[
30x - 16 = 0 \quad \Rightarrow \quad x = \frac{16}{30} = \frac{8}{15}
\]

Thus, \( x = \frac{8}{15} \) is a potential inflection point.

### 4. **Behavior at Infinity**:
As \( x \to \infty \), the dominant term is \( 5x^3 \), so:

\[
\lim_{x \to \infty} f(x) = \infty, \quad \lim_{x \to -\infty} f(x) = -\infty
\]

### 5. **Key Features**:
- **Critical points**: \( x = \frac{2}{3}, x = \frac{2}{5} \)
- **Inflection point**: \( x = \frac{8}{15} \)
- **Increasing/Decreasing**: Analyze based on the sign of \( f'(x) \) in intervals around the critical points.
- **Concavity**: Analyze based on the sign of \( f''(x) \).

Would you like a detailed explanation of any specific part or further analysis?

### Related Questions:
1. What are the local maxima and minima of \( f(x) \)?
2. Can you determine intervals where \( f(x) \) is concave up or concave down?
3. How do you graph \( f(x) \) based on its critical points and inflection points?
4. What is the significance of the second derivative in terms of concavity?
5. How does the cubic term \( 5x^3 \) influence the end behavior of \( f(x) \)?

### Tip:
Always check the sign of the first and second derivatives to understand how a function behaves (increasing/decreasing and concavity).

Explore a detailed solution for the cubic function f(x) = 5x^3 - 8x^2 + 4x. This analysis covers first and second derivatives, critical points, concavity, and inflection points. Suitable for students in Grades 11-12 or early college calculus.

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