[Math] Secant and Bisection Method

Question

What would be the example of a function for which a Secant Method fails but Bisection Method converges (to the root). In particular, if we are checking the interval $[a,b]$, then starting points for the Secant Method are $a$ and $b$.

Answer 1

Try to find a continuously differentiable function with the following properties:

• $f(a)$ and $f(b)$ have opposite signs and
• $f'(\xi) = 0$ for a $\xi \in [a,b]$

The first point ensures that the bisection methods converges. Whereas if $f'(\xi)=0$, the secant method can fail. See these lecture notes (page 101) for an example.