OldSmokey, thanks for pointing out the difference in propeller and jet engine aircrafts. I was referring to single engine propeller aircraft. I found a site here that offers a very very good explanation, like yours. It is a very good site:
http://www.eaa1000.av.org/technicl/p...s/perfspds.htm
To quote:
To maximize the endurance, we want to maximize the amount of time that we can stay in the air. In order to do this, we must minimize the fuel flow. Since the fuel flow is proportional to the power required, the fuel flow will be minimized at the point where the power required is a minimum. The speed corresponding to the bottom of the power required curve is the speed for maximum endurance
and,
To minimize the pounds of fuel per nautical mile, we can minimize the ratio of power over velocity. Looking at the power required chart, a line from the origin to any point on the curve has the slope of power over velocity (P/V). As you trace a line from the origin to each point on the curve, the slope will be a minimum when the line is tangent to the power required curve. Therefore, the maximum range airspeed occurs where a line from the origin is tangent to the power required curve. This also corresponds to the minimum point on the thrust required curve (drag polar).
and for Carson's Speed,
Unfortunately the maximum range airspeed is generally a lot slower than most people wish to fly. After all, you built an airplane to get places fast. Since we are also interested in getting places fast, we must consider speed. So consider a parameter of fuel flow per knot (Fuel Flow/knot). This would tell us how much fuel per hour we are burning for each knot of velocity. The optimum speed would then be the speed where this parameter is a minimum. Mathematically, the derivative with respect to velocity would equal zero