Question: The differentiable function $f$ is shown below on the domain $(-9, 9)$. Determine which statement is true regarding $f'(-5)$, $f'(0)$, and $f'(5)$.
1. **State the problem:** We need to compare the values of the derivative $f'$ at $x = -5$, $x = 0$, and $x = 5$ based on the graph of $f$.
2. **Recall the meaning of $f'(x)$:** The derivative $f'(x)$ represents the slope of the tangent line to the graph of $f$ at the point $x$.
3. **Analyze $f'(-5)$:** At $x = -5$, the graph has a minimum point. The slope of the tangent line at a minimum is $0$, so $f'(-5) = 0$.
4. **Analyze $f'(0)$:** At $x = 0$, the graph is decreasing (the curve is going down), so the slope is negative. Therefore, $f'(0) < 0$.
5. **Analyze $f'(5)$:** At $x = 5$, the graph is increasing (the curve is going up), so the slope is positive. Therefore, $f'(5) > 0$.
6. **Compare the values:** Since $f'(0) < 0$, $f'(-5) = 0$, and $f'(5) > 0$, the order is:
$$f'(0) < f'(-5) < f'(5)$$
7. **Conclusion:** The true statement is:
$f'(0) < f'(-5) < f'(5)$