Question

The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity:
f(v)=(v^3-78v^2+5200v)/150000.

a. What is the optimal velocity which minimizes the fuel consumption of the vehivle in gallons Per mile?

b. What is the critical number?

Answer 1

solve for the derivative of f(v). Set f'(v) = 0 to find your critical values. Put them on a number line. Solve for values less than and greater than those critical values. To find the minimum you need to have a value whose values to the left of it give you (-'s) i.e, the slopes are -. and to the left of it give you (+'s). This is a minimum critical value, which when you sub into the function f(v) will give you the velocity you are looking for. Part b, is the pt that you found. It looks like your derivative is parabolic and should have two critical values on your f'(v) number line. Choose the easiest numbers on the intervals in between your critical numbers to find out how the slopes change. - to + is a min pt, + to - is a max point. They are called local extrema and the points at which they occur are called inflection points. You probably have to use the quadratic formula to find your zeroes on the derivative.