$V_{md}$ is the speed where the rate of sink is the lowest. So if you add full power, it should be the speed with greatest rate of climb, namely $V_y$.
But we also know $V_{md}<V_{bg}$ according to this speed polar graph as a general rule 
If I connect the inequalities, I have $V_y=V_{md}<V_{bg}$. In the particular Cherokee I'm trained in, $V_y$ is 75 knots and $V_{bg}$ is 69 knots. So where in the reasoning am I wrong? Why isn't $V_y$ equal to $V_{md}$?
Vy includes engine/prop efficiency factors, so it can not be directly compared to glide data. Vbg is the best speed for DISTANCE, which is what you want to make your emergency landing site if you have an "engine out". Vmd will give you a lower sink rate, but you will not glide as far.
Interestingly, if you have to glide INTO the wind, best glide speed will be FASTER than Vbg to make up for loss of groundspeed. Best glide speed there for is the optimal ratio of altitude given for ground gained.
