f0 = -(x^6 - 3/2 x^4 + x^2);
$a = Factor[((f0 /. x -> p) - (f0 /. x -> q))/(p - q)];
doubleT = {Factor[((D[f0, x] /. x -> p) - (D[f0, x] /. x -> q))/(
p - q)] == 0,
(D[f0, x] /. x -> p) - $a == 0,
p < q};
dtsol = Solve[doubleT, {p, q}, Reals];
dtsol = Select[
dtsol,
AllTrue[D[f0, {x, 2}] /. {{x -> p}, {x -> q}} /. #, Negative] &];
asol = {a -> $a} /. dtsol;