To identify the intervals where the function \( y = 0.4x^3 - 4 \) is increasing, decreasing, or constant, we can follow these steps:

### 1. **Find the derivative:**

The derivative of a function tells us its rate of change, which we can use to determine where the function is increasing or decreasing.

\[
y = 0.4x^3 - 4
\]
The derivative is:
\[
\frac{dy}{dx} = \frac{d}{dx}(0.4x^3 - 4) = 1.2x^2
\]

### 2. **Critical points:**

Critical points occur where the derivative equals zero or is undefined.

\[
1.2x^2 = 0
\]

Solving this equation, we get:
\[
x = 0
\]

So, the critical point is \( x = 0 \).

### 3. **Determine the behavior of the function:**

Now, we'll check the sign of \( 1.2x^2 \) on both sides of the critical point \( x = 0 \).

- For \( x < 0 \), \( 1.2x^2 > 0 \) (positive)
- For \( x > 0 \), \( 1.2x^2 > 0 \) (positive)

Since the derivative is positive for all \( x \neq 0 \), the function is **increasing** on both intervals \( (-\infty, 0) \) and \( (0, \infty) \).

### 4. **Constant region:**

The derivative is zero at \( x = 0 \), which means the function has a **horizontal tangent** at this point, but it's not constant over any interval.

### Conclusion:
- The function is **increasing** on the intervals \( (-\infty, 0) \) and \( (0, \infty) \).
- The function has a **horizontal tangent** at \( x = 0 \), but it's still increasing overall.

---

Would you like further details or have any questions?

Here are 5 questions to expand on the topic:
1. How do we determine the concavity of the function?
2. What does the second derivative tell us about this function?
3. How can we find inflection points of the function?
4. What happens to the function as \( x \) approaches positive or negative infinity?
5. How do we interpret the graph of this cubic function?

**Tip:** When finding intervals of increase or decrease, always check the derivative and analyze its sign across the critical points.

Learn how to find the intervals of increase for the function y = 0.4x^3 - 4. This detailed solution walks through using the derivative to identify critical points and analyze the behavior of the function across different intervals. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation