For the intermediate value theorem why do you think it is necessary for the signs of f(a) and f(b) to be different in order to guarantee there is a zero between a and b.
If it is the case that the signs of f(a) and f(b) are the same, does that mean the function f(x) cannot have a zero between a and b? Explain
The intermediate value theorem only says something about regions of functions that are continuous. So picture your function as a continuous, smooth, no-nonsense function between two points a and b. Now picture that function having a zero between a and b. If f(a) is positive and f(b) is negative (or vice versa), since the function is continous, it MUST pass through y = 0 to get from f(a) to f(b). If f(a) and f(b) are the same sign, there is no guarantee that the function passes through y = 0 for this reason. For clarity, look at the function f(x) = x^2. This function is never negative, so f(a) and f(b) are always >= 0. If you look at the interval [-1,1] (thus a = -1, b = 1), you’ll notice that f(a) and f(b) are both positive. But f(0) = 0! So 0 is in our interval, but the intermediate value theorem can’t predict that there is a zero on our interval. If you are looking at the interval [1,5], for example (thus a = 1, b = 5), you’ll notice that f(a) and f(b) are both positive, and there is no zero in that interval. Therefore, if f(a) and f(b) are the same sign, there may or may not be a zero on the interval. Either way, the intermediate value theorem cannot predict it because a continous function is not required to pass through the x axis to go from one point to another of the same sign. I recommend you look at some functions and apply the intermediate value theorem on different intervals to get a feel for this yourself.
Hope this helps,