Question about stopping criteria for bisection method. I was just told the following:
The stopping criterion in the bisection method is $|b − a| < \epsilon$. However for most other algorithms it is is $|f(x)| < \epsilon$. These two stopping criteria are within a factor of approximately $|f'(x^*)|$ assuming $f$ is smooth and $\epsilon$ is small.
I'm confused by what "within a factor of approximately $|f'(x^*)|$" means and how to go about showing this.

Near 0 we can say that $f(\frac{b+a}{2})\approx df(x)$ for $dx=b-a$. So the bigger $|f'(x*)|$ (where $x*$ is true solution) is, the lesser $dx$ has to be.
