The function f ( x ) = x 4 − 9 x 3 + 20 x 2 f\left(x\right)=x^4-9x^3+20x^2 has a root of multiplicity 2 2 at x = 0 x=0 . Apply both the traditional Newton's method and the modified Newton's method starting from x 0 = 0.5 x_0=0.5 to find x 3 x_3 . Which method provides a closer approximation to the root? Modified Newton's method (for multiplicity 2 2 ): x n + 1 = x n − 2 f ( x n ) f ′ ( x n ) x_{n+1}=x_{n}-\frac{2f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}

The modified Newton's method gives a closer approximation to the root at x = 0 x=0 x = 0 after three iterations.

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4. Applications of Derivatives

Differentials

4. Applications of Derivatives

Differentials: Study with Video Lessons, Practice Problems & Examples

117PRACTICE PROBLEM

ANSWERS OPTIONS

The modified Newton's method gives a closer approximation to the root at $x=0$ after three iterations.

The traditional Newton's method gives a closer approximation to the root at $x=0$ after three iterations.

The traditional and modified Newton's methods result in equal root approximations after three iterations.

Neither the traditional nor the modified Newton's method gives an accurate root approximation after three iterations.