Step 1 :Convert the total time from minutes to hours: \(34 \text{ minutes} = 0.5666666666666667 \text{ hours}\)
Step 2 :Calculate the average speed by dividing the total distance by the total time: \(\frac{6.2 \text{ miles}}{0.5666666666666667 \text{ hours}} = 10.941176470588236 \text{ miles per hour}\)
Step 3 :Since the runner's speed was zero at the start and end of the race, and the runner's speed must have increased from zero to some maximum value and then decreased back to zero, the runner must have been running at exactly 10 miles per hour at least twice in the race.
Step 4 :According to the Mean Value Theorem (MVT), there must be at least one point in the race where the runner's speed was exactly equal to the average speed.
Step 5 :\(\boxed{\text{The correct choice is A. The average speed is approximately 10.9 miles per hour. By the Mean Value Theorem, the speed was exactly 10.9 miles per hour at least once. All speeds between 0 and 10.9 miles per hour were reached. Because the initial and final speed was 0 miles per hour, the speed of 10 miles per hour was reached at least twice in the race.}}\)