Root Location Theorem

    The intermediate value theorem can be used to estimate roots of a given equation For example, suppose and then by the intermediate value thereom, there exists at least one number in the open interval such that

Proposition (Root Location Theorem) If is continuous on the closed interval and if and have opposite signs, then for at least one in the open interval

Example (Root Location Theorem) Use the root location theorem to find a root of a function on an interval. Verify that has at least one solution on the interval .

    Solution. Notice that the function is continuous on and in particular is continuous on Then, since and , the intermediate value theorem says that there is at least one real number between and such that   Here's the graph:

Example (Root Location Theorem) Use the root location theorem to find a root of a function on an interval. Show that has at least one solution on the interval

    Solution. The function is continuous on and so it is continuous on Since and by the Root Location Proposition has at least one root on Here's the graph:
    

Example (Root Location Theorem) Use the root location theorem to find a root of a function on an interval. Show that has at least one solution on the interval

    Solution. The function is continuous on and so it is continuous on Since and by the Root Location Proposition has at least one root on Here's the graph:
    

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Root Location Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/root-location-theorem.html