(*1*)
intervalHalving[f_, a_, b_, eps_] := 
 Module[{xm = (a + b)/2, L = b - a, x1 = a + L/4, x2 = b - L/4}, 
  While[Abs[L] < eps, x1 = a + L/4;
     x2 = b - L/4;
     If[(f /. x -> x1) < f /. x -> xm, b = xm;
      xm = x1,
      If[(f /. x -> x2) < (f /. x -> xm),
        a = xm;
        xm = x2,
        a = x1;
        b = x2];
      ];
     L = b - a]
    Print["Optimal solution for function f: ", xm];
  ]




intervalHalving[(x - 3)^2, -1, 5, 0.1]
intervalHalving[x^4 + 5 x^2 - 10 x, -2, 4, 0.1]

Optimal solution for function f: 2

Optimal solution for function f: 1

(*2*)
goldenPartition[f_, a_, b_, eps_] := 
 Module[{p = 0.382, xm, w1 = a + p*(b - a), w2 = a + (1 - p)*(b - a), 
   fw1 = f /. x -> w1, fw2 = f /. x -> w2}, While[Abs[b - a] < eps,
   w1 = a + p*(b - a);
   w2 = a + (1 - p) (b - a);
   If[fw1 > fw2,
    a = w1,
    If[fw2 > fw1,
      b = w2,
      a = w1;
      b = w2];
    ];
   ];
  xm = (a + b)/2;
  Print["Approximate minimal point of function f is: ", xm];
  ]


goldenPartition[(x - 3)^2, -1, 5, 0.5]
goldenPartition[x^4 + 5 x^2 - 10 x, -2, 4, 0.5]


Approximate minimal point of function f is: 2

Approximate minimal point of function f is: 1

(*3*)
fibonacciNumbers[f_, a_, b_, n_] := Module[{k = 1, x1, x2, xm},
  L[k_] := Fibonacci[n - k + 1]/Fibonacci[n + 1];
  While[k == n, k = k + 1;
   x1 = a + L[k];
   x2 = b - L[k];
   If[x1 < x2, ,
    temp = x1;
    x1 = x2;
    x2 = temp];
   If[(f /. x -> x1) > (f /. x -> x2),
    a = x1,
    If[(f /. x -> x2) > (f /. x -> x1), b = x2, a = x1;
      b = x2];
    ];
   ];
  xm = (a + b)/2;
  Print["Approximate minimal point of function f: ", xm];
  ]


fibonacciNumbers[(x - 3)^2, -1, 5, 4]
fibonacciNumbers[x^4 + 5 x^2 - 10 x, -2, 4, 4]


Approximate minimal point of function f: 2

Approximate minimal point of function f: 1