Consider the value ????̂ that maximizes a function L(????). This exercise motivates the Newton–Raphson method by focusing on the single-parameter case.

a. Using L′

(????̂) = L′

(????(0)) + (????̂ − ????(0))L′′(????(0)) + ⋯, argue that for an initial approximation ????(0) close to ????̂, approximately 0 = L′

(????(0)) + (????̂ −

????(0))L′′(????(0)). Solve this equation to obtain an approximation ????(1) for ????̂.

b. Let ????(t) denote approximation t for ????̂, t = 0, 1, 2,…. Justify that the next approximation is

????(t+1) = ????(t) − L′

(????(t)

)∕L′′(????(t)

).