Why is the gradient of Energy vs. wavenumber(k) zero at the edge of Brillouin zones?
The x axis on the plot is proportional to momentum.  The k-zone boundary in your diagram, is the point with zero momentum (no net motion of a carrier.);  So, in the plot you show, that's why it's the minimum point as all other points have a net motion greater than it. You will notice, that the graph is symmetrical about the k=0 point in the kronig-penny model.  The gradient is zero because of that symmetry. From basic calculus, when you take the derivative of a function, it is always zero at an extremium.