The Zero-Derivative Theorem

    The following theorem is a partial converse to the statement that the derivative of a constant is 0.

Proposition (Zero-Derivative Theorem) Let be a function that is continuous on and differentiable on If for all in then is constant on

    Proof. If and are different points in then by the Mean Value Theorem there exists a in such that
    


By hypothesis and so Since and were chosen arbitrarily, is a constant function on

Example (Zero-Derivative Theorem) Consider Notice that for all in the domain, but is not a constant. Doe this example contradict the Zero-Derivative Theorem?

    Solution. No it does not, rather it shows that the assumptions of the zero-derivative theorem are necessary.

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Zero Derivative Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/zero-derivative-theorem.html