This problem is based on the Intermediate Value Theorem and a concept called the Mean Value Theorem from calculus. Here's how we can analyze it:

Step-by-Step Explanation:

1. Average Speed Calculation:

   • The runner completes a 6.2-mile race in 34 minutes.
   • To find the average speed over the entire race, we calculate: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{6.2 \text{ miles}}{34 \text{ minutes}} = \frac{6.2 \times 60}{34} \approx 10.94 \text{ miles per hour}$

   So, the runner’s average speed over the course of the race is approximately 10.94 mph.

2. Mean Value Theorem (MVT):

   • The Mean Value Theorem for continuous functions tells us that if a function is continuous and differentiable over an interval, there exists at least one point in the interval where the instantaneous rate of change (or instantaneous speed, in this case) equals the average rate of change.
   • Since the runner's speed is continuous throughout the race, there must be at least one point where his speed is exactly 10.94 mph.

3. Intermediate Value Argument:

   • The problem asks why the runner must have been running at exactly 10 mph at least twice.
   • Since his speed at the start was presumably zero (starting from rest) and also zero at the finish line (as stated in the problem), his speed must have varied throughout the race, crossing different values.
   • Because his average speed was around 10.94 mph, he would have crossed 10 mph at least once on his way up to his peak speed and at least once again on his way back down toward zero at the finish.

4. Conclusion:

   • By the Intermediate Value Theorem, since he started below 10 mph, went above it, and returned to zero, he must have hit exactly 10 mph at least twice during the race.

Summary

Therefore, the runner must have been running at exactly 10 mph at least twice during the race due to the continuous nature of his speed and the application of the Intermediate Value Theorem.

Would you like a more detailed breakdown, or have any specific questions?

Follow-up Questions

1. How does the Mean Value Theorem apply to speed and distance?
2. Why is the Intermediate Value Theorem crucial in this context?
3. Could there be more than two points where the runner’s speed was exactly 10 mph?
4. How would the problem change if the runner's speed at the start was not zero?
5. What assumptions about the runner's motion are implicit in using calculus here?

Tip:

The Mean Value Theorem can help in many physics and motion problems to identify specific points where certain conditions must hold.