Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Which of the following statements about the bisection, secant, and Newton methods is/are correct?
For a starting interval $[a, b]$, the bisection method fails if $f(a) > f(b)$.
The order of convergence of the bisection method is 2.
On every iteration of the Newton method, one needs to Taylor expand the function to second order around the current estimate of the root.
If $f$ contains exactly one root in the starting interval $[a, b]$ with $f(a)f(b) < 0$, then the secant method will converge to this root.
The order of convergence of the Newton method is 1.
The secant method cannot be applied with a starting interval $[a, b]$ such that $f(a) = f(b)$.