Exploring Extremes of Inequalities in Variables x and y

GeoGebra Commands & Parameters

1. TextToUnicode(LaTeX[a]) → list_x
2. TextToUnicode(LaTeX[b]) → list_y
3. Intersect(g, eq1) → D
4. Intersect(g, eq2) → C
5. Intersect(f, eq2) → B
6. Intersect(f, eq1) → A
7. Polygon(D, C, B, A) → q1, c, d, b_1, a_1
8. Textfield(x_l) → InputBox1
9. Textfield(x_u) → InputBox2
10. Textfield(y_l) → InputBox3
11. Textfield(y_u) → InputBox4
12. Max(x_l, x_u) → max_x
13. Min(x_l, x_u) → min_x
14. Min(y_l, y_u) → min_y
15. Max(y_l, y_u) → max_y
16. Min({(min_x * min_y), (max_x * min_y), (max_y * min_x), (max_x * max_y)}) → ans_c
17. If(x_u > 0 ∧ x_l < 0 ∨ x_u ≟ 0 ∨ x_l ≟ 0, true, false) → minxsqzero
18. If(y_u > 0 ∧ y_l < 0 ∨ y_u ≟ 0 ∨ y_l ≟ 0, true, false) → minysqzero
19. If(minysqzero, {min_x / min_y, max_x / min_y, min_x / max_y, max_x / max_y, max_x / 0}, {min_x / min_y, max_x / min_y, min_x / max_y, max_x / max_y}) → listxdivy
20. Max(listxdivy) → ans_d
21. Min(y_l - 3, -3) → yscrmin
22. Max(y_u + 3, 3) → yscrmax
23. Min(x_l - 3, -3) → xscrmin
24. Max(x_u + 5, 5) → xscrmax
25. If(minxsqzero, 0, Min[min_x^(2), max_x^(2)]) → minsqx
26. Max(min_x^(2), max_x^(2)) → maxsqx
27. If(minysqzero, 0, Min[min_y^(2), max_y^(2)]) → minsqy
28. Max(min_y^(2), max_y^(2)) → maxsqy
29. Max(listysqminusx) → ans_e
30. If(minysqzero, {max_y^(2) / min_x, min_y^(2) / min_x, minsqy / min_x, max_y^(2) / max_x, min_y^(2) / max_x, minsqy / max_x}, {max_y^(2) / min_x, min_y^(2) / min_x, max_y^(2) / max_x, min_y^(2) / max_x}) → listysqdivx
31. If(minysqzero, {LaTeX["\frac{" + "(" + max_y + ")²" + "}{" + min_x + "}"], LaTeX["\frac{" + "(" + min_y + ")²" + "}{" + min_x + "}"], LaTeX["\frac{" + minsqy + "}{" + min_x + "}"], LaTeX["\frac{" + "(" + max_y + ")²" + "}{" + max_x + "}"], LaTeX["\frac{" + "(" + min_y + ")²" + "}{" + max_x + "}"], LaTeX[("\frac{" * minsqy) + "}{" + max_x + "}"]}, {LaTeX["\frac{" + "(" + max_y + ")²" + "}{" + min_x + "}"], LaTeX["\frac{" + "(" + min_y + ")²" + "}{" + min_x + "}"], LaTeX["\frac{" + "(" + max_y + ")²" + "}{" + max_x + "}"], LaTeX["\frac{" + "(" + min_y + ")²" + "}{" + max_x + "}"]}) → listtxtysqdivx
32. Min(listysqdivx) → ans_f
33. If(IsInRegion[A, ylessthanx] ∧ IsInRegion[C, ylessthanx] ∨ IsInRegion[A, ygreaterthanx] ∧ IsInRegion[C, ygreaterthanx], true, false) → sqyminusxnotzero
34. Segment(A, C) → i
35. If(Defined[Intersect[i, h]], Intersect[i, h], (99, 99)) → E
36. If(sqyminusxnotzero, {(max_y - max_x)^(2), (max_y - min_x)^(2), (min_y - max_x)^(2), (min_y - min_x)^(2)}, {floor(y(E)) - floor(x(E)), (max_y - max_x)^(2), (max_y - min_x)^(2), (min_y - max_x)^(2), (min_y - min_x)^(2)}) → listsqofyminusx
37. Min(listsqofyminusx) → ans_g
38. If(sqyminusxnotzero, {LaTeX["((" + max_y + ")-(" + max_x + "))²"], LaTeX["((" + max_y + ")-(" + min_x + "))²"], LaTeX["((" + min_y + ")-(" + max_x + "))²"], LaTeX["((" + min_y + ")-(" + min_x + "))²"]}, {LaTeX["((" + (floor(y(E))) + ")-(" + (floor(x(E))) + "))^2"], LaTeX["((" + max_y + ")-(" + max_x + "))²"], LaTeX["((" + max_y + ")-(" + min_x + "))²"], LaTeX["((" + min_y + ")-(" + max_x + "))²"], LaTeX["((" + min_y + ")-(" + min_x + "))²"]}) → listtxtsqofyminusx
39. Min(listyminusxcubed) → ans_h
40. If(¬minysqzero, {min_x^(2) - min_y^(2), max_x^(2) - min_y^(2), min_x^(2) - max_y^(2), max_x^(2) - max_y^(2)}, {min_x^(2) - 0, max_x^(2) - 0}) → listxsqminusysq
41. Max(listxsqminusysq) → ans_i
42. If(¬minysqzero, {LaTeX["(" + min_x + ")^2-(" + min_y + ")^2"], LaTeX["(" + max_x + ")^2-(" + min_y + ")^2"], LaTeX["(" + min_x + ")^2-(" + max_y + ")^2"], LaTeX["(" + max_x + ")^2-(" + max_y + ")^2"]}, {LaTeX[("(" * min_x) + ")²" + "-(" + 0 + ")^2"], LaTeX[("(" * max_x) + ")²" + "-(" + 0 + ")^2"]}) → listtxtxsqminusysq
43. Min(listxcubedminusysq) → ans_j
44. If(minxsqzero, {LaTeX["\frac{" + min_y + "}{2(^-" + 0 + ")}"], LaTeX["\frac{" + max_y + "}{2(^-" + 0 + ")}"], LaTeX["\frac{" + max_y + "}{2(" + 0 + "^+)}"], LaTeX["\frac{" + min_y + "}{2(" + 0 + "^+)}"], LaTeX["\frac{" + min_y + "}{2(" + min_x + ")}"], LaTeX["\frac{" + max_y + "}{2(" + min_x + ")}"], LaTeX["\frac{" + min_y + "}{2(" + max_x + ")}"], LaTeX["\frac{" + max_y + "}{2(" + max_x + ")}"]}, {LaTeX["\frac{" + min_y + "}{2(" + min_x + ")}"], LaTeX["\frac{" + max_y + "}{2(" + min_x + ")}"], LaTeX["\frac{" + min_y + "}{2(" + max_x + ")}"], LaTeX["\frac{" + max_y + "}{2(" + max_x + ")}"]}) → listtxtydiv2x
45. If(minxsqzero, {((-min_y)) / 0, ((-max_y)) / 0, max_y / 0, min_y / 0, min_y / ((2 * min_x)), max_y / ((2 * min_x)), min_y / ((2 * max_x)), max_y / ((2 * max_x))}, {min_y / ((2 * min_x)), max_y / ((2 * min_x)), min_y / ((2 * max_x)), max_y / ((2 * max_x))}) → listydiv2x
46. Max(listydiv2x) → ans_k