Consider the value that maximizes a function L( ). Let 0 denote an initial guess.

Consider the value β̂ that maximizes a function L( β). Let β⁰ denote an initial guess.

a. Using L’( β̂.) = L’( β⁽⁰⁾ + (β̂ – β⁽⁰⁾ L”(β⁽⁰⁾) + ..., argue that for β⁽⁰⁾ close to β̂, approximately 0 = L’( β⁽⁰⁾) + (β̂ – β⁽⁰⁾) L’’(β⁽⁰⁾).

Solve this equation to obtain an approximation β⁽¹⁾ for β̂.

b. Let β^(t) denote approximation t for β̂, t = 0, 1, 2, ... Justify that the next approximation is β^{(t + 1)} = β^(t) – L’(β^(t))/L”(β^(t)).