Intermediate Value Theorem

    The intermediate value property is known as: a continuous function on a closed bounded interval must take on all values between and

Proposition (Intermediate Value Theorem) If is a continuous function on the closed interval and is some number strictly between and , then there exists at least one number on the open interval such that

Example (Application of the Intermediate Value Theorem) The population (in thousands) of a colony of bacteria minutes after the application of a toxin is given by the function



    (a) When does the colony die out?

    (b) Show that at some time between and the population is
    
    Solution.  Since



and



we know that is coninuous at and thus, also for all   Here is a graph of



(a) The colony dies out when which means Therefore, the colony dies out in 8 minutes and 15 seconds.

(b) Since and is continuous on the intermediate value theorem yields at least one number between 2 and 7 such Therefore, there is some time between and such that the population is 9,000.  

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Intermediate Value Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/intermediate-value-theorem.html