Given the function f, prime, of, x, equals, x, cubed, minus, 15, x, commaf 
′
 (x)=x 
3
 −15x, determine all intervals on which ff is decreasing.

To determine the intervals on which \( f(x) \) is decreasing, we need to analyze the behavior of the derivative \( f'(x) = x^3 - 15x \). Here's the step-by-step approach:

1. **Find Critical Points:**
   Set \( f'(x) = 0 \) to find critical points, as these points can indicate changes in the increasing or decreasing behavior.
   \[
   x^3 - 15x = 0
   \]
   Factor out \( x \):
   \[
   x(x^2 - 15) = 0
   \]
   This gives:
   \[
   x = 0 \quad \text{or} \quad x = \pm \sqrt{15}
   \]

2. **Analyze the Sign of \( f'(x) \):**
   To determine the intervals where \( f(x) \) is decreasing, check the sign of \( f'(x) \) on each interval defined by the critical points \( x = -\sqrt{15}, 0, \sqrt{15} \).

   Let's analyze \( f'(x) = x^3 - 15x \) over each interval:

   - For \( x < -\sqrt{15} \): Choose a test point (e.g., \( x = -4 \)). 
     - \( f'(-4) = (-4)^3 - 15(-4) = -64 + 60 = -4 \) (negative).
   
   - For \( -\sqrt{15} < x < 0 \): Choose a test point (e.g., \( x = -1 \)).
     - \( f'(-1) = (-1)^3 - 15(-1) = -1 + 15 = 14 \) (positive).

   - For \( 0 < x < \sqrt{15} \): Choose a test point (e.g., \( x = 1 \)).
     - \( f'(1) = (1)^3 - 15(1) = 1 - 15 = -14 \) (negative).

   - For \( x > \sqrt{15} \): Choose a test point (e.g., \( x = 4 \)).
     - \( f'(4) = (4)^3 - 15(4) = 64 - 60 = 4 \) (positive).

3. **Determine Intervals of Decrease:**
   From the sign analysis:
   - \( f'(x) < 0 \) for \( x \in (-\infty, -\sqrt{15}) \) and \( x \in (0, \sqrt{15}) \).
   
   Therefore, \( f(x) \) is **decreasing** on the intervals:
   \[
   (-\infty, -\sqrt{15}) \quad \text{and} \quad (0, \sqrt{15})
   \]

---

Would you like a breakdown of any step or a graph of \( f(x) \) to illustrate? Here are some related questions to deepen understanding:

1. How would we find intervals where \( f(x) \) is increasing?
2. What if \( f(x) \) were given as \( x^4 - 15x \) instead—how would the intervals change?
3. What role does the second derivative \( f''(x) \) play in determining the concavity of \( f(x) \)?
4. How can we confirm the behavior of \( f(x) \) at the critical points themselves?
5. What effect would changing the constant in \( f(x) \) have on these intervals?

**Tip:** Always test points within intervals to confirm behavior, especially with higher-order polynomials, where behavior can change multiple times.

Given the function f'(x) = x^3 - 15x, determine all intervals on which f is decreasing.

Determine the intervals on which the function f(x) is decreasing by analyzing its derivative, f'(x) = x^3 - 15x. This problem involves finding critical points, testing intervals, and using the first derivative test. Ideal for students in Grades 11-12 learning calculus and derivatives.

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