Extremum of the graph of a function has integer coordinates

Question

Graph of the fuction $y=\dfrac{(x-26)(x+9)}{(x+14)(x+19)}$ has maximum point and minimum point are (-16,-49) and (-4,-1) whose coordinates are integer numbers. I am trying to find four integer numbers a, b, c, d so that graph of the function $f(x) = \dfrac{(x+a)(x+b)}{(x+c)(x+d)}$ has two extremum $(x_1, f(x_1))$, $(x_2, f(x_2))$ where $x_1$, $x_2$, $f(x_1)$, $f(x_2)$ are four integer numbers. I tried

Clear["Global`*"];
f[x_] = ((x + a) (x + b))/((x + c) (x + d));
sol = Solve[f'[x] == 0, x] // Simplify;
x1 = x /. sol[[1]]
x2 = x /. sol[[2]]
f[x1] // FullSimplify
f[x2] // FullSimplify
Table[{IntegerQ[x1] && IntegerQ[x2] &&  IntegerQ[f[x1]] && 
   IntegerQ[f[x2]]}, {-30 <= a <= 30 && -30 <= b <= 30 && -30 <= c <= 
    30 && -30 <= d <= 30 && a > b  && c > d}]

I can not get the result. How can I get the results?

Answer 1

My answer is based on another answer on this site which I do not remember its link. I am sorry the author of this answer.

PS. My answer is based on this answer

I use your result

Clear["Global`*"];
f[x_] = ((x + a) (x + b))/((x + c) (x + d));
sol = Solve[f'[x] == 0, x] // Simplify;
r1 = x /. sol[[1]]
r2 = x /. sol[[2]]
r3 = f[r1] // FullSimplify
r4 = f[r2] // FullSimplify

and input

Clear["Global`*"];
grid = Tuples[DeleteCases[Range[-30, 30], 0], 4];
cpicked = 
  With[{IntegerQ = FractionalPart[#] == 0 &}, 
   Compile[{{m, _Real, 1}}, 
    Module[{a = m[[1]], b = m[[2]], c = m[[3]], d = m[[4]], r1, r2, 
      r3, r4}, 
     If[(a - c) (b - c) (a - d) (b - d) < 0, 0, 
      r1 = -((-a b + Sqrt[(a - c) (b - c) (a - d) (b - d)] + c d)/(a +
             b - c - d));
      r2 = (a b + Sqrt[(a - c) (b - c) (a - d) (b - d)] - c d)/(a + 
          b - c - d);
      r3 = (b (c + d) + a (-2 b + c + d) - 
          2 (Sqrt[(a - c) (b - c) (a - d) (b - d)] + c d))/(c - d)^2;
      r4 = (2 Sqrt[(a - c) (b - c) (a - d) (b - d)] - 2 c d + 
          b (c + d) + a (-2 b + c + d))/(c - d)^2;
      a > b && c > d && a != b != c != d && IntegerQ[r1] && 
        IntegerQ[r2] && IntegerQ[r3] && IntegerQ[r4] // Boole]], 
    RuntimeAttributes -> {Listable}]];
(ans = Pick[grid, cpicked@grid, 1])

{{5, -30, 15, 10}, {6, -29, 16, 11}, {7, -28, 17, 12}, {8, -27, 18, 13}, {9, -26, 19, 14}, {10, -25, 20, 15}, {11, -24, 21, 16}, {12, -23, 22, 17}, {13, -22, 23, 18}, {14, -21, 24, 19}, {15, -20, -25, -30}, {15, -20, 25, 20}, {16, -19, -24, -29}, {16, -19, 26, 21}, {17, -18, -23, -28}, {17, -18, 27, 22}, {18, -17, -22, -27}, {18, -17, 28, 23}, {19, -16, -21, -26}, {19, -16, 29, 24}, {20, -15, -20, -25}, {20, -15, 30, 25}, {21, -14, -19, -24}, {22, -13, -18, -23}, {23, -12, -17, -22}, {24, -11, -16, -21}, {25, -10, -15, -20}, {26, -9, -14, -19}, {27,-8, -13, -18}, {28, -7, -12, -17}, {29, -6, -11, -16}, {30, -5, -10, -15}}