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this video is going to be so fetched lets talk about the mean value theorem so first a preliminary result known as rolls theorem we got f of x its a function its continuous on the interval a which includes its endpoints this is a connected compact set its differentiable on the interior were not going to address derivatives at the end points but at least except at the end points it has a derivative and our final assumption is that f of a equals f b the value of the function at the endpoints is equal then the conclusion of the theorem is that theres some point in between a and b where the derivative is 0. now heres a picture to sort of show what were given and what were concluding so here is our interval from a to b the function just has to be continuous from left to right and have a derivative on the interior also the height at the left endpoint is equal to the height at the right end point so if i label this as f b and i label this as f of a those are at the same height now
