Approximating Zeros of a Continuous Function

You are given a continuous function $f$ on an interval $[a, b]$ with $f(a)$ and $f(b)$ of opposite sign. Describe an algorithm to approximate a zero of $f$ (i.e., a point $x^{*}$ where $f(x^{*}) = 0$). 1. Explain the simplest guaranteed approach and analyze its convergence. 2. Describe at least one faster alternative and explain the trade-off. 3. What method would you recommend in practice, and why?