Escape velocity of a satellite

When you look at the formula for escape velocity, you can see that it only depends on the distance from the massive body and that body's mass.
In other words, it doesn't matter what you do around that mass, just where you are.

 

Escape velocity is a speed, not a velocity. The easiest way to attain that velocity is for a vehicle to thrust in the direction it is already going. The minimum delta V needed to escape at some point is the difference between the current velocity at that point and the escape velocity at that point. It is this value you want to minimize, not the escape velocity.

 

That depends on your definition of "optimal". For rockets the optimal point is characterized by the highest velocity:

http://en.wikipedia.org/wiki/Oberth_effect

 

You used the wrong expression for orbital velocity.

A good place to start is the vis-viva equation. Unfortunately, you'll get no joy here if you differentiate the resulting delta V with respect to r. The problem is there's nothing that constrains r in either the vis-viva equation or the escape velocity equation.

You'll need something such as the orbital equation of motion, ##r=\frac{a (1-e^2)}{1-e\cos\theta}##. Now you should get something useful. In particular, you should find that extrema occur at apofocus and perifocus. So now it's just a matter of determining which is best, which is worst.