Clear[x, det, g]
det = Det[Table[Sin[(n - m)^2*x], {n, 1, 6}, {m, 1, 6}]];
g[y_?NumberQ] := det /. x -> y
To get the min can use NMinimize.
NMinimize[{g[x], Pi/2 <= x <= Pi}, x]
Out[201]= {-15.6808053249, {x -> 1.94853409545}}
This will plot it.
Plot[g[x], {x, Pi/2, Pi}, PlotRange -> All]
