The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: $$f(v) = \frac{v^3-74 v^2 + 5100 v}{150000}$$ 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 $v$:
$g(v) =$
Hint for the above: Assume the vehicle is moving at constant velocity $v$. 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 $v=$.
To verify that $g(v)$ has a minimum at this critical number we compute the second derivative $g''(x)$ and find that its value at the critical number is , a positive number.