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    Properties of Continuous Functions : Intermediate Value Theorem

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    Suppose that functions f,g : [a,b] -> R are continuous, satisfy f(a) <= g(a)
    and f(b) >= g(b). Then there exists a real number c in [a,b] such that f(c) = g(c).

    Label the statement as true or false. If it is true, prove it. If not, give an example of why it is false and if possible, correct it to make it true.

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    Solution Preview

    Dear student,

    This one can be done as follows: We have f(x) and g(x), x is an element of [a,b]. f and g are continuous and real-valued on [a,b]. We have f(a)<=g(a), f(b)>=g(b).
    There are 4 possibilities:
    1. f(a)=g(a), f(b)>g(b)
    2. f(a)<g(a), f(b)=g(b)
    3. f(a)=g(a), f(b)=g(b)
    4. f(a)<g(a), ...

    Solution Summary

    Properties of Continuous Functions and the Intermediate Value Theorem are investigated.