f(x)=8x^3-72x^2 list the maximal open intervals over which f(x) is decreasing

To find the intervals over which the function \( f(x) = 8x^3 - 72x^2 \) is decreasing, we first need to find the critical points by following these steps:

### Step 1: Find the first derivative of \( f(x) \).
The first derivative \( f'(x) \) gives us information about the slope of the function, which helps in determining where the function is increasing or decreasing.

\[
f'(x) = \frac{d}{dx}(8x^3 - 72x^2) = 24x^2 - 144x
\]

### Step 2: Set the first derivative equal to zero to find critical points.
Critical points occur where \( f'(x) = 0 \) or is undefined. Let's solve for \( x \) by setting \( f'(x) = 0 \):

\[
24x^2 - 144x = 0
\]

Factor out the common term \( 24x \):

\[
24x(x - 6) = 0
\]

This gives us two solutions:

\[
x = 0 \quad \text{or} \quad x = 6
\]

### Step 3: Determine the sign of \( f'(x) \) on the intervals defined by the critical points.
The critical points divide the real number line into three intervals: \( (-\infty, 0) \), \( (0, 6) \), and \( (6, \infty) \).

We test the sign of \( f'(x) \) within each interval by choosing a test point from each interval:

- For \( (-\infty, 0) \), pick \( x = -1 \):

\[
f'(-1) = 24(-1)^2 - 144(-1) = 24 + 144 = 168 \quad (\text{positive})
\]

- For \( (0, 6) \), pick \( x = 3 \):

\[
f'(3) = 24(3)^2 - 144(3) = 24(9) - 432 = 216 - 432 = -216 \quad (\text{negative})
\]

- For \( (6, \infty) \), pick \( x = 7 \):

\[
f'(7) = 24(7)^2 - 144(7) = 24(49) - 1008 = 1176 - 1008 = 168 \quad (\text{positive})
\]

### Step 4: Conclusion
The function \( f(x) \) is decreasing where \( f'(x) < 0 \), which occurs on the interval \( (0, 6) \).

### Final Answer:
The function \( f(x) = 8x^3 - 72x^2 \) is decreasing on the open interval \( (0, 6) \).

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Would you like more details on this or have any other questions?

Here are 5 related questions:
1. How can we use the second derivative to determine concavity of \( f(x) \)?
2. What are the increasing intervals of the function \( f(x) \)?
3. How do we classify critical points as local maxima or minima?
4. Can you explain how to sketch the graph of \( f(x) \) based on this analysis?
5. What happens to the function as \( x \to \infty \)?

**Tip:** When solving for intervals of increasing or decreasing behavior, always use the first derivative to test the sign in different intervals.

f(x) = 8x^3 - 72x^2, list the maximal open intervals over which f(x) is decreasing

Discover how to find the decreasing intervals for the function f(x) = 8x^3 - 72x^2. This problem uses key calculus concepts such as derivatives and critical points, and is suitable for students in Grades 11-12.

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