Rolle's Theorem

in this video we're gonna focus on a roll stir and we're gonna go over some example problems associated with it so what is rolle's theorem what's the main idea behind it there are three conditions that must be met so let's say if we have a function f of X this function has to be continuous on the

closed interval a B and it has to be differentiable on the open interval a B and also f of a must equal F of B if these three conditions are met then we have this particular conclusion there's gonna be some number C such that F prime of C is zero so basically there's gonna

be a horizontal tangent at some point C between a and B so let's draw a picture that illustrates this so let's say if we have a graph that looks like this

so is the function continuous on the closed interval what would you say yes it is there's no holes no vertical asymptotes no jumps in that curve now is the function differentiable on the open interval a B yes it is notice that we have a nice smooth curve there's no sharp terms no jumps nothing like that now

there's F of a equal F of B what would you say it's the third condition met now notice that F of a and F of B they have the same Y value so therefore they're equal to each other if these three conditions are met then there's gonna be a horizontal tangent somewhere between a

and B and we can see where it is it's at at this point so that's Point C and that Point C we can see that the slope of the horizontal tangent is zero so thus f prime of C is equal to zero and that's the main idea behind Rolle's Theorem so let me give you another example another