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: $(-\infty, -\sqrt{15}) \quad \text{and} \quad (0, \sqrt{15})$

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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.