Optimization one variable, how to calculate extreme value theorem

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belgiumrob

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Jan 18, 2016
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Hi everyone,

So, I have the following problem, which i solved for a big part. Now I am stuck with my answer. Is it a maximum or minimum, is it global or local? and what about the extreme value theorem? Thanks in advance for your feedback! :)

An international transport company wants to convey cargo by truck at the lowest cost.The most important shipping costs originate from the diesel price and the truck driver’swage. Both these costs, however, depend on the speed of the transport : The higher thespeed the higher the diesel consumption will be, but the lower the labour cost is. If thedistance is traveled at an average speed of v km/h , then the diesel consumption is estimatedat 10 + v4liter per 100 kilometer. At which (average) speed is the cruise control ofthe truck to be set to keep the total costs as low as possible, if diesel costs 1.20 euro perliter, the driver’s wage is 19.20 euro per hour, a distance of 1000 km has to be covered,and the maximal allowed speed for this type of transport is 90 km/h?

So, I defined the variables:
v=km/h
consumption: 10+v/4 liter / 100km
wage: 19,20 /hour
distance 1000km
max V: 90km/h

Total cost function is then: (19,2*1000/v) + 10(1,2(10 + v/4)), where I want to look for the lowest cost, with restrictions: V>0 and V<90
If I am correct, then the function can be simplified to: 19200v^-1 + 3V + 120

Then I take the derivative f'(v) = -19200v^-2 +3
This I can solve as v^2=6400 or v=80 or v=-80.

The restriction says that only v=80 is valid.

Until here I am fine. But then? how do I proceed from here to know whether v=80 is a local/global minimum/maximum, and how to calculate further the extreme values?

I hope my question is clear, and hoping for your reply/replies,

Thanks,
belgiumrob
 
belgiumrob said:
Hi everyone,

So, I have the following problem, which i solved for a big part. Now I am stuck with my answer. Is it a maximum or minimum, is it global or local? and what about the extreme value theorem? Thanks in advance for your feedback! :)

An international transport company wants to convey cargo by truck at the lowest cost.The most important shipping costs originate from the diesel price and the truck driver’swage. Both these costs, however, depend on the speed of the transport : The higher thespeed the higher the diesel consumption will be, but the lower the labour cost is. If thedistance is traveled at an average speed of v km/h , then the diesel consumption is estimatedat 10 + v4liter per 100 kilometer. At which (average) speed is the cruise control ofthe truck to be set to keep the total costs as low as possible, if diesel costs 1.20 euro perliter, the driver’s wage is 19.20 euro per hour, a distance of 1000 km has to be covered,and the maximal allowed speed for this type of transport is 90 km/h?

So, I defined the variables:
v=km/h
consumption: 10+v/4 liter / 100km
wage: 19,20 /hour
distance 1000km
max V: 90km/h

Total cost function is then: (19,2*1000/v) + 10(1,2(10 + v/4)), where I want to look for the lowest cost, with restrictions: V>0 and V<90
If I am correct, then the function can be simplified to: 19200v^-1 + 3V + 120

Then I take the derivative f'(v) = -19200v^-2 +3
This I can solve as v^2=6400 or v=80 or v=-80.

The restriction says that only v=80 is valid.

Until here I am fine. But then? how do I proceed from here to know whether v=80 is a local/global minimum/maximum, and how to calculate further the extreme values?

I hope my question is clear, and hoping for your reply/replies,

Thanks,
belgiumrob
Click to expand...

Use the second derivative rule to determine whether it's a local max or min.

Since your function is continuous when v>0, consider what happens when speed is at the allowable extremes (ie 90 and >0).
 
Thanks Harry_the_cat, ofcourse the 2nd derivative, i forgot indeed.

Thanks for pointing out
 
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