Rolle's Theorem

    The Extreme Value Theorem guarantees the existence of a maximum and minimum value of a continuous function on a closed bounded interval. The next theorem is called Rolle's Theorem and it guarantees the existence of an extreme value on the interior of a closed interval, under certain conditions
    Basically Rolle's theorem states that if a function is differentiable on an open interval, continuous at the endpoints, and if the function values are equal at the endpoints, then it has at least one horizontal tangent. Of course if the function is constant this is automatically true for all points in the interval. So the point is that Rolle's theorem guarantees us at least one point in the interval where there will be a horizontal tangent. Rolle's theorem is a special case of the Mean Value Theorem for when the values of the function are the same at the endpoints of the interval.

Proposition (Rolle's Theorem) Let be a function that is continuous on , differentiable on , and   Then there exists at least one number in such that

    Proof. If is a constant function, then the statement is true; in fact for all in If for some in then by the Extreme Value Theorem, attains its absolute maximum value somewhere in the open interval But precisely at this we have, If for some in then by the Extreme Value Theorem, attains its absolute minimum value somewhere in the open interval But precisely at this we have,
    

Example (Rolle's Theorem) Verify Rolle's theorem for on

    Solution. Notice that is continuous and differentiable for all real numbers. Also, and therefore Rolle's theorem applies and so there is at least one in such that We can find it by solving In fact we find two, namely



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Example (Application of Rolle's Theorem) Prove that the equation has exactly one real root.

    Solution. Since the function is a polynomial it is continuous and differentiable for all real numbers. Thus, the Intermediate Value Theorem and Rolle's Theorem applies. Since and by the Intermediate Value Theorem there is a in such that Therefore, the equation has at least one solution.  To prove that for only one we assume that there are two roots namely, and ; and we prove that this can not happen. Thus, assume and are solutions, that is with Then by Rolle's Theorem there exists a in such that Notice that so that in fact such a can not exist. Therefore, there can not be and in fact the equation has exactly one real root.

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Rolles Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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