Consider the following equation. 3x4 − 8x3 + 6 = 0, [2, 3] (a) Explain how we know that the given equation must have a root in t
he given interval. Let f(x) = 3x4 − 8x3 + 6. The polynomial f is continuous on [2, 3], f(2) = < 0, and f(3) = > 0, so by the Intermediate Value Theorem, there is a number c in (2, 3) such that f(c) = . In other words, the equation 3x4 − 8x3 + 6 = 0 has a root in [2, 3]. (b) Use Newton's method to approximate the root correct to six decimal places.
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