(*This file is used to explore the bounds on the power function and its derivatives.
    For fixed exponent, these functions are monotonic in their arguments.  So, we
    only need to understand what happens at the two endpoints of the interval
    of interest.  The two endpoints are 1/4 and 1/2 + 2^{-9}.  When the exponent
   s=e/2 is at most 1, the various maxima occur at the left endpoints.  This is the
   case of interest in the paper.*)

s=1/4

f0[s_]:=Power[x,s];
f1[s_]:=s Power[x,s-1];
f2[s_]:=s (s-1) Power[x,s-2];
f3[s_]:=s (s-1) (s-2) Power[x,s-3];

(*Numerically computed max for the first, second, and third derivatives.  An
actual proof would require something more rigorous, but we only care about
the case s=1/2,1.  For these cases, we can just compute directly.*)

g1:=(
SOL1=NSolve[D[f1[s],s]==0,s];
f1[SOL1[[1,1,2]]])

g2:=(
SOL2=NSolve[D[f2[s],s]==0,s];
{f2[SOL2[[1,1,2]]],f2[SOL2[[2,1,2]]]})

g3:=(
SOL3=NSolve[D[f3[s],s]==0,s];
{f3[SOL3[[1,1,2]]],f3[SOL3[[2,1,2]]],f3[SOL3[[3,1,2]]]})