For the bisection (halving the interval) approximation method, if the root is a turning point (double root) on the x axis, this method would fall apart right?
Since there is a root despite having no change in sign on either side.
So for questions that says, e.g.
"show that there are no roots, or if there are roots, between x = 3 and x = 4, getting two same or opposing signs respectively on either side isn't enough and we'll need to differentiate to confirm?" (unless we know what the graph looks like, of course).
Also, what does Maths In Focus means when they say
"If we halve the interval several times, the approximation to the root will USUALLY, but NOT always, become MORE accurate.