Finding the maxima or minima of band-edges?

[LaVela]

I'm working on a homework assignment and I'm completely stuck on this last problem. I'm not even sure what ∇_KE=0 even means to begin with. I understand the difference between indirect and direct bandgap but I'm just confused on how to find the maxima and minima to determine this. Any help would be greatly appreciated.

 

[ZapperZ]

This really should be posted in the HW/Coursework forum.

First of all, don't you remember from your calculus class that a maximum and minimum will have zero gradient? ∇KE=0 is exactly that, except that this is obviously a gradient in 3D k-space. So find the k values for each one of them where the gradient is zero. If the k values you found for each band is the same, then the max and min occurs at the same k values and so, you have a direct band gap. If not, you have an indirect gap.

Zz.