Question: How can I insert L'(2) =0 and L''(2) > 0 in this list?

I have a list:
mylist := [x^4 + (-4*m - 7)*x^3 + (m + 4)*x^2 + (3*m - 5)*x, x^4 + (-4*m - 7)*x^3 + (m + 5)*x^2 + (5*m - 7)*x, x^4 + (-4*m - 7)*x^3 + (m + 5)*x^2 + (7*m - 5)*x, x^4 + (-4*m - 7)*x^3 + (2*m + 5)*x^2 + (3*m - 5)*x, x^4 + (-4*m - 7)*x^3 + (3*m + 1)*x^2 + (7*m - 10)*x, x^4 + (-4*m - 5)*x^3 + (2*m + 1)*x^2 + (7*m - 9)*x]

I use

L := map~(normal, mylist);

and get. 

L := [x^4 - (4*m + 7)*x^3 + (m + 4)*x^2 + (3*m - 5)*x, x^4 - (4*m + 7)*x^3 + (m + 5)*x^2 + (5*m - 7)*x, x^4 - (4*m + 7)*x^3 + (m + 5)*x^2 + (7*m - 5)*x, x^4 - (4*m + 7)*x^3 + (2*m + 5)*x^2 + (3*m - 5)*x, x^4 - (4*m + 7)*x^3 + (3*m + 1)*x^2 + (7*m - 10)*x, x^4 - (4*m + 5)*x^3 + (2*m + 1)*x^2 + (7*m - 9)*x].

I use seq to list L[i] and diff(L[i])

[seq([L[i], diff(L[i], x)], i = 1 .. nops(L))];

and get

[[x^4 - (4*m + 7)*x^3 + (m + 4)*x^2 + (3*m - 5)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(m + 4)*x + 3*m - 5], [x^4 - (4*m + 7)*x^3 + (m + 5)*x^2 + (5*m - 7)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(m + 5)*x + 5*m - 7], [x^4 - (4*m + 7)*x^3 + (m + 5)*x^2 + (7*m - 5)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(m + 5)*x + 7*m - 5], [x^4 - (4*m + 7)*x^3 + (2*m + 5)*x^2 + (3*m - 5)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(2*m + 5)*x + 3*m - 5], [x^4 - (4*m + 7)*x^3 + (3*m + 1)*x^2 + (7*m - 10)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(3*m + 1)*x + 7*m - 10], [x^4 - (4*m + 5)*x^3 + (2*m + 1)*x^2 + (7*m - 9)*x, 4*x^3 - 3*(4*m + 5)*x^2 + 2*(2*m + 1)*x + 7*m - 9]]

How can I insert L'(2), L''(2) and solve the systems L'(2) = 0 and L''(2) > 0 to get the solutions m?
like this
[seq([L[i], diff(L[i], x), solve([L'(2) = 0,L''(2)>0],m) ], i = 1 .. nops(L))]

I also tried
[seq([L[i], diff(L[i], x), eval(diff(L[i], x), x = 2), solve([eval(diff(L[i], x), x = 2) = 0], m)], i = 1 .. nops(L))]

to obtain 
[[x^4 - (4*m + 7)*x^3 + (m + 4)*x^2 + (3*m - 5)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(m + 4)*x + 3*m - 5, -41 - 41*m, {m = -1}], [x^4 - (4*m + 7)*x^3 + (m + 5)*x^2 + (5*m - 7)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(m + 5)*x + 5*m - 7, -39 - 39*m, {m = -1}], [x^4 - (4*m + 7)*x^3 + (m + 5)*x^2 + (7*m - 5)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(m + 5)*x + 7*m - 5, -37 - 37*m, {m = -1}], [x^4 - (4*m + 7)*x^3 + (2*m + 5)*x^2 + (3*m - 5)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(2*m + 5)*x + 3*m - 5, -37 - 37*m, {m = -1}], [x^4 - (4*m + 7)*x^3 + (3*m + 1)*x^2 + (7*m - 10)*x, 4*x^3 - 3*(4*m + 7)*x^2 + 2*(3*m + 1)*x + 7*m - 10, -58 - 29*m, {m = -2}], [x^4 - (4*m + 5)*x^3 + (2*m + 1)*x^2 + (7*m - 9)*x, 4*x^3 - 3*(4*m + 5)*x^2 + 2*(2*m + 1)*x + 7*m - 9, -33 - 33*m, {m = -1}]]