In the setting of Examples 2 and 3, let r denote the speed along the road, and h denote the speed along the highway.
(a) Show that the travel–time function T(x) has a critical point at x = 30/√(h/r)² − 1 and explain why this indicates that if r ≥ h there is no critical point.
(b) Explain why there cannot be a critical point at x = 0, but depending on the speeds, the critical point can be arbitrarily close to 0.

Example 2

Example 3

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EXAMPLE 2 Minimizing Travel Time Your task is to build a road joining the small town of Calverton to Route 1 to enable drivers to reach Capital City in the shortest time (Figure 3). How should this be done if the speed limit is 60 km/hour on the road and 110 km/h on Route 1? The perpendicular distance from Calverton to Route 1 is 30 km, and Capital City is 50 km down Route 1.