To solve a problem related to Rolle’s Theorem in the context of Traffic Flow Optimization, let's outline a scenario and apply the theorem.

Problem Setup:

Consider a straight road between two points, A and B, over a 10 km stretch. The traffic flow rate $f(t)$ (vehicles per minute) at time $t$ (in minutes) varies smoothly during the time period from 9:00 AM to 9:30 AM.

Assume the following conditions:

• The traffic flow at 9:00 AM and 9:30 AM is the same (100 vehicles per minute).
• The flow is continuous and differentiable during the period from 9:00 AM to 9:30 AM.

We are tasked with showing that there must exist some time $t \in (0, 30)$, during which the traffic flow rate is neither increasing nor decreasing. That is, we need to show that at some point between 9:00 AM and 9:30 AM, the rate of change of traffic flow is zero (i.e., the traffic flow reaches a local maximum or minimum).

Step-by-Step Solution:

Step 1: Applying Rolle’s Theorem

Rolle’s Theorem states that if a function $f(t)$ satisfies the following conditions on a closed interval $[a, b]$:

1. $f(t)$ is continuous on $[a, b]$,
2. $f(t)$ is differentiable on $(a, b)$, and
3. $f(a) = f(b)$,

Then there exists at least one point $c \in (a, b)$ such that the derivative $f'(c) = 0$.

Step 2: Verifying Conditions

Let’s define $f(t)$ as the traffic flow rate at time $t$, where $t$ is measured in minutes from 9:00 AM.

• Continuity: Traffic flow is continuous in real-world situations since flow doesn’t suddenly jump or drop but changes smoothly over time. Hence, $f(t)$ is continuous.
• Differentiability: Traffic flow is differentiable because it changes smoothly, meaning there are no sudden breaks or jumps in the flow rate, and we can compute its rate of change.
• Equal values at endpoints: The problem states that the traffic flow rate at 9:00 AM and 9:30 AM is the same, i.e., $f(0) = f(30) = 100$ vehicles per minute.

Thus, all conditions of Rolle’s Theorem are satisfied.

Step 3: Conclusion

By Rolle’s Theorem, there must exist some time $t = c$ in the interval $(0, 30)$ such that the derivative $f'(c) = 0$, which means the rate of change of traffic flow at this time is zero. In practical terms, this indicates that at some point between 9:00 AM and 9:30 AM, the traffic flow reached a local maximum or minimum and the flow rate was neither increasing nor decreasing.

Interpretation in Traffic Flow:

The point where the derivative is zero could represent a period of congestion relief (if it’s a local maximum) or a period before traffic congestion worsens (if it’s a local minimum).

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Would you like further details on any part of the solution? Here are some related questions to think about:

1. How can Rolle’s Theorem help predict traffic jams in a more complex traffic model?
2. What other conditions must be considered for a traffic flow model to ensure real-world applicability?
3. How does Rolle’s Theorem compare with the Mean Value Theorem in traffic flow analysis?
4. Can Rolle’s Theorem be applied if the traffic flow rate is not differentiable everywhere?
5. What kind of data would we need to apply these theorems to real-world traffic?

Tip: In real-world applications, the existence of a critical point (like in Rolle’s Theorem) helps planners to anticipate possible traffic congestion or bottlenecks and devise better traffic signal timings or rerouting plans.