Find or derive the equation for power for your particular setup. If you are not given the equation, calculate it from work and time, or from force and velocity. If you do not have these variables, calculate them from their respective equations. Keep going backwards, calculating simpler variables based on the information you do have.
Decide the variable you want to change in order to minimize power. For example, if the device involves a spring, and you want to know what kind of spring to use, you will minimize power with respect to spring constant (k). If instead, you want to know how far to pull the spring, you will minimize power with respect to displacement (x).
Take the derivative of the equation for power with respect to the variable you chose in step 2.
Set the derivative equal to zero, and solve for the variable from step 2. This answer is called the critical point.
Plug this value into the equation for power to find the extreme value of power. Note that this value might be the maximum or the minimum power, so it is important to proceed to the next step.
Double-check that your answer is the absolute minimum by plugging in "endpoints" -- the lowest and highest possible values of the variable step 2. Sometimes, endpoints will yield a lesser power than that found by setting the derivative equal to zero. In such case, make sure that the endpoint is a physically reasonable answer, and use it instead.
Often, solving for the variable that yields minimum power will give several possible critical points. In this case, plug each value -- as well as the endpoints -- to discover which value yields the absolute minimum power (the lowest answer).