EXAMPLE 8 Show that there is a root of the equation 8x3 11x2 3x 2 = 0 between 1 and 2. SOLUTION Let f(x) = 8x3 11x2 3x 2 = 0. We are looking for a solution of the given equation, that is, a number c between 1 and 2 such that f(c) = . Therefore we take a = , b = , and N = in the Intermediate Value Theorem. We have f(1) = 8 11 3 2 = 2 < 0 and f(2) = 64 44 6 2 = 24 > 0. Thus f(1) < 0 < f(2); that is N = 0 is a number between f(1) and f(2). Now f is continuous since it is a polynomial, so that the Intermediate Value Theorem says there is a number c between (smaller) and (larger) such that f(c) = . In other words, the equation 8x3 11x2 3x 2 = 0 has at least one root c in the open interval . In fact, we can locate a root more precisely by using the Intermediate Value Theorem again. Since f(1.2) = 0.416 < 0 and f(1.3) =