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    Intermediate value theorem

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    Define Intermediate-Value theorem
    Using this theorem, show that there is a root of P(x)=x^3+x^2+x-1 in the closed interval [0,1].

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    Solution Preview

    P(x) is continuous. Since P(0)=-1 and P(1)=2 there must be a number in such that (Since 0 is between -1 and 2). We can do even better. We have

    Therefore there is a root between ...

    Solution Summary

    This defines Intermediate-Value theorem and uses it in a proof.