Hello everyone! I came across this problem in my homework and I've been having a lot trouble with it. If anyone could tell me how to solve it or at least where to start that would be amazing.;)

thank you very much!

For a fish swimming at a speed v relative to the water, the energy expenditure per until time is proportional to v^3. It is believed that migrating fish try to minimize the total energy required a fixed distance. If the fish are swimming against a current u (u<v) then the time required to swim a distance L is L / (u-v) and the total energy required to swim the distance is given by:

E(v)= (av^3) ( L/ v-u)

where a is a proportionality constant.

a) Determine the value of v that minimizes E.

I won't solve this for you, but here's the approach to use:

First, take the derivative of E(v) with respect to v, and find for what value of v the derivative is equal to 0. This value will give either a minimum or a maximum for E(v). To determine which it is, you calculate the second derivative at that point - if the 2nd derivative is a positive value, then the function is "bath tub shaped," and you have a local minimum. If the 2nd derivative is positive, then the function is an inverted bath tub, and you have a local maximum. (Hint - you'll find that for your problem that the first derivative does indeed give a minimum.) Lastly, you should check the boundary values, to ensure that there isn't something funny going on - in this case, what is E(v) for v = u (the min that v can be) and for v = infinity, and how do they compare to the value of the local min you found above?

All this assumes you now how to take first and second derivatives - post back if you have questions about that. Good luck.