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# (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton's method to approximate the root correct to six decimal places.$3x^4 - 8x^3 + 2 = 0$, $[2, 3]$

## a) In other words, the equation $3 x^{4}-8 x^{3}+2=0$ has a root in [2,3]b) $x \approx 2.630020$

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okay, Part A. We can use the intermediate value theorem because we know there must be at least 10 on the interval. 2 to 3, because the function takes on every value between negative 14 29 including zero zero's betweennegative 14 and 29. Therefore, for part B, the first thing we can do is take the derivative. Okay, we know X one is approximately 2.5. It's between two and three, therefore X of two 2.5 minutes off 2.5 over a prime of 2.5 is two point sex sex X three. Using the same Newton's method, we get 2.63 Ark's four. We get 2.63 Next. Five, we get 2.63 So Acts is 2.6300 to 0.

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