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# Prove the second derivative of a function with a given property must have a zero

Consider a function which is continuous everywhere on an interval and has a second derivative everywhere on the open interval . Assume the chord joining two points and on the graph of the function intersects the graph of the function at a point with . Prove there exists a point such that .

Proof. Let be the equation of the line joining and . Then define a function Since and intersect at the values , and this means By our definition of then we have Further, since and are continuous and differentiable on , we apply Rolle’s theorem twice: first on the interval and then on the interval . These two applications of Rolle’s theorem tells us there exist points and such that Then, we apply Rolle’s theorem for a third time, this time to the function on the interval to conclude that there exists a such that . Then, since we know (since ). So, Thus, for some 