Question

Consider the following equation: 3x^4 - 8x^3 + 3 = 0, [2, 3] (a) Explain how we know that the given equation must have a root in the given interval. Let f(x) = 3x^4 - 8x^3 + 3. Answer for part A: The polynomial f is continuous on [2, 3], f(2) = -13 < 0, and f(3) = 30 > 0, so by the Intermediate Value Theorem, there is a number c in (2, 3) such that f(c) = 0. In other words, the equation 3x^4 - 8x^3 + 3 = 0 has a root in [2, 3]. (b) Use Newton's method to approximate the root correct to six decimal places.

          Consider the following equation:
3x^4 - 8x^3 + 3 = 0, [2, 3]
(a) Explain how we know that the given equation must have a root in the given interval. Let f(x) = 3x^4 - 8x^3 + 3.
Answer for part A: The polynomial f is continuous on [2, 3], f(2) = -13 < 0, and f(3) = 30 > 0, so by the Intermediate Value Theorem, there is a number c in (2, 3) such that f(c) = 0. In other words, the equation 3x^4 - 8x^3 + 3 = 0 has a root in [2, 3].
(b) Use Newton's method to approximate the root correct to six decimal places.

Added by Tom-S Q.

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