The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: [math] What is the optimal velocity which minimizes the fuel consumption of the vehicle in gallons PER MILE?
To solve this problem, we need to minimize the following function of [math]:
[math]
Hint for the above: Assume the vehicle is moving at constant velocity [math]. How long will it take to travel 1 mile? How much gas will it use during that time?
We find that this function has one critical number at [math] .
To verify that [math] has a minimum at this critical number we compute the second derivative [math] and find that its value at the critical number is , a positive number.