Using the Intermediate Value Theorem

Use the Intermediate Value Theorem to show that \begin{equation*} f(x)=x^{3}+x^{2}-x-2 \end{equation*}

has a zero between \(1\) and \(2\).

Solution Since \(f\) is a polynomial, it is continuous on the closed interval \([1,2]\). Because \(f(1)=-1\) and \(f(2)=8\) have opposite signs, the Intermediate Value Theorem states that \(f(c)=0\) for at least one number \(c\) in the interval \((1,2)\). That is, \(f\) has at least one zero between \(1\) and \(2.\) Figure 33 shows the graph of \(f\) on a graphing utility.