Consider the following. −2x 5 + 9x 4 − 7x 3 − 12x = 0, [3, 4] (a) Explain how we know that the given equation must have a solution in the given interval. Let f(x) = −2x 5 + 9x 4 − 7x 3 − 12x. The...

Consider the following.

−2x⁵ + 9x⁴ − 7x³ − 12x = 0, [3, 4]

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

Let
f(x) = −2x
⁵
+ 9x
⁴
− 7x
³
− 12x. The polynomialf is continuous on [3, 4],

f(3) = ??? > 0,

and

f(4) = ??? <>

so by the Intermediate Value Theorem, there is a number
c
in (3,4) such that

f(c) =??? .

In other words, the equation

−2x⁵ + 9x⁴ − 7x³ − 12x = 0

has a solution in [3, 4].

(b)Use Newton's method to approximate the solution correct to six decimal places.

x = ???

Jun 05, 2022

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