https://file.aiquickdraw.com/mathbot/file/17273149684633aa903up.jpg

The problem is asking for the values of \( x \) where the function \( f(x) = x^3 - 3x^2 - 15 \) is decreasing.

To solve this, we first need to find the derivative of \( f(x) \) and analyze where it is negative (since a function decreases when its derivative is negative).

### Step 1: Derive \( f(x) \)
The function is:
\[
f(x) = x^3 - 3x^2 - 15
\]
We find the first derivative \( f'(x) \):
\[
f'(x) = 3x^2 - 6x
\]

### Step 2: Factor the derivative
Now, factor the derivative expression:
\[
f'(x) = 3x(x - 2)
\]

### Step 3: Determine where \( f'(x) \) is negative
To find where \( f'(x) \) is negative, analyze the critical points by solving \( f'(x) = 0 \):
\[
3x(x - 2) = 0
\]
This gives the critical points \( x = 0 \) and \( x = 2 \).

Next, we use these critical points to determine the intervals where \( f'(x) \) is negative. Test the sign of \( f'(x) \) in the following intervals:

- For \( x < 0 \) (e.g., \( x = -1 \)):
  \[
  f'(-1) = 3(-1)((-1) - 2) = 3(-1)(-3) = 9 > 0
  \]
  So, \( f'(x) > 0 \) in this interval (function is increasing).

- For \( 0 < x < 2 \) (e.g., \( x = 1 \)):
  \[
  f'(1) = 3(1)(1 - 2) = 3(1)(-1) = -3 < 0
  \]
  So, \( f'(x) < 0 \) in this interval (function is decreasing).

- For \( x > 2 \) (e.g., \( x = 3 \)):
  \[
  f'(3) = 3(3)(3 - 2) = 3(3)(1) = 9 > 0
  \]
  So, \( f'(x) > 0 \) in this interval (function is increasing).

### Step 4: Conclusion
The function decreases when \( 0 < x < 2 \). Thus, the correct answer is:

**B. \( 0 < x < 2 \)**

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the intervals where a function is increasing or decreasing?
2. What is the significance of the derivative being zero at critical points?
3. Can a function have both increasing and decreasing intervals? How?
4. How would you determine the concavity of a function using the second derivative?
5. What role do critical points play in sketching the graph of a function?

**Tip**: Always test the sign of the derivative in each interval to determine whether a function is increasing or decreasing.

Fungsi f(x) = x^3 - 3x^2 - 15 turun untuk semua x yang memenuhi ....

Find the intervals where the function f(x) = x^3 - 3x^2 - 15 is decreasing. This problem focuses on finding the derivative and applying the first derivative test. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation