To find the intervals of time when the particle's speed is increasing, we need to analyze the behavior of the function $x(t)$, given its graph.

1. Key Insight: Speed is the absolute value of velocity, $|v(t)|$. The speed increases when $v(t)$ and $a(t)$ (the acceleration) have the same sign.

2. Steps to Solve:

   • Identify velocity: $v(t) = x'(t)$, which is the slope of the graph of $x(t)$.
   • Identify acceleration: $a(t) = x''(t)$, which corresponds to the concavity of the graph. When the graph of $x(t)$ is concave up, $a(t) > 0$; when concave down, $a(t) < 0$.

3. Inflection Points: These points, marked with green circles, indicate where $a(t)$ changes sign.

Analysis

Observing the graph:

• The inflection points (marked by green circles) occur roughly around $t \approx 6$, $t \approx 13$, and another around $t \approx 15$.
• Between these inflection points, the concavity (sign of $a(t)$) changes, which helps determine when $v(t)$ and $a(t)$ have the same sign.

Intervals Where Speed is Increasing

1. From $t = 0$ to $t \approx 6$: The slope (velocity) is positive, and the graph is concave up (acceleration is positive). Speed increases.

2. From $t \approx 13$ to $t \approx 15$: The slope (velocity) is negative, and the graph is concave down (acceleration is negative). Speed increases.

Thus, the intervals where the particle's speed is increasing are approximately: $(0, 6) \quad \text{and} \quad (13, 15)$

Would you like further details on how each interval was determined?

Related Questions:

1. What conditions determine when speed is decreasing?
2. How would the intervals change if the concavity points were different?
3. What is the relationship between inflection points and speed changes?
4. How does one determine intervals of constant speed using this graph?
5. Can this approach apply to other types of graphs representing motion?

Tip: When analyzing motion graphs, always differentiate between velocity and speed, as speed depends on the magnitude, while velocity depends on direction.