The following result is a particular case of a theorem proved by Professor David C

Chapter 12, Problem 50

(choose chapter or problem)

The following result is a particular case of a theorem proved by Professor David C. Kurtz in The American Mathematical Monthly [vol. 99 (1992), pp. 259263]. Suppose we have a cubic equation a3x3 a2x2 a1x a0 0 in which all of the coefficients are positive real numbers. Furthermore, suppose that the following two inequalities hold. Then the cubic equation has three distinct real roots. (a) Check that these inequalities are valid in the case of the equation 2x3 8x2 7x 1 0. This implies that the equation has three distinct real roots. Use a graphing utility to verify this and to estimate each root to the nearest one hundredth. (b) Follow part (a) for the equation 3x3 40x2 100x 30 0. (c) Use a graphing utility to demonstrate that the graph of y 6x3 15x2 11x 2 has three distinct x-intercepts. Thus, the equation 6x3 15x2 11x 2 0 has three distinct real roots. Now check that the condition 4a1a3 fails to hold in this case. Explain why this does not contradict the result from Professor Kurtz stated above. 51. (a) Use a