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EXAMPLE 8 Show that there is a root of the equation 3x36x2+3x2=0 between 1 and 2. SOLUTION Let f(X) = 3x3 6x2 + 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) = C]. Therefore we take a = C] , b = C] , and N = |:] in the Intermediate Value Theorem. We have f(1)=36+32=2o. Thus f(1) 0 a root must lie between |:] (smaller) and Cl (larger). A calculator gives, by trial and error, f(1.63) = 0.059159 0. So a root lies in the open interval