Consider the following equation.

3x⁴ 8x³ + 5 = 0, [2, 3]

(a) Explain how we know that the given equation must have a root in the given interval.Let

f(x) = 3x⁴ 8x³ + 5.

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 + 5 = 0

has a root in [2, 3]. (b) Use Newton's method to approximate the root correct to six decimal places.