a. Explanation Theorem used: Newton’s method: Newton’s iterative method to compute approximate roots of a function f ( x ) = 0 is given as follows: 1. Choose an initial approximation value x 0 which is close to the root. 2. For n th known value, the next value is obtained as, x n + 1 = x n − f ( x n ) f ′ ( x n ) Here, n varies as 0 , 1 , 2 , 3 , … and f ′ is the derivative of the function f such that f ′ ( x n ) ≠ 0 . Calculation: The derivative of the function f ( x ) = 4 x 4 − 4 x 2 is, f ′ ( x ) = 16 x 3 − 8 x Note that, for the given function f ( x ) = 4 x 4 − 4 x 2 the Newton’s formula is as follows. That is, x n + 1 = x n − 4 x n 4 − 4 x n 2 16 x n 3 − 8 x n Use the initial condition as x 0 = − 2 . Substitute x n = − 2 in f ( x n ) and simplify further. f ( − 2 ) = 4 ( − 2 ) 4 − 4 ( − 2 ) 2 = 4 ( 16 ) − 4 ( 4 ) = 64 − 16 = 48 Substitute x n = − 2 in f ′ ( x n ) and simplify further. f ′ ( x n ) = 16 ( − 2 ) 3 − 8 ( − 2 ) = 16 ( − 8 ) + 16 = − 128 + 16 = − 112 Substitute x n = − 2 in x n + 1 = x n − f ( x n ) f ′ ( x n ) and simplify further b. To determine The zeros of the function f ( x ) = 4 x 4 − 4 x 2 by using the starting values x 0 = − 0.5 and x 0 = 0.25 , lying ( − 21 7 , 21 7 ) . c. To determine The zeros of the function f ( x ) = 4 x 4 − 4 x 2 by using the starting values x 0 = 0.8 and x 0 = 2 , lying ( 2 2 , ∞ ) . d. To determine The zeros of the function f ( x ) = 4 x 4 − 4 x 2 by using the starting values x 0 = − 21 7 and x 0 = 21 7 .

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Chapter 4.6, Problem 29E

To determine

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=−2 and x0=−0.8, lying (−∞,−22).

To determine

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=−0.5 and x0=0.25, lying (−217,217).

To determine

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=0.8 and x0=2, lying (22,∞).

To determine

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=−217 and x0=217.

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Chapter 4.6, Problem 28E

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The zeros of the function f ( x ) = 4 x 4 − 4 x 2 by using the starting values x 0 = − 2 and x 0 = − 0.8 , lying ( − ∞ , − 2 2 ) .

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The zeros of the function f(x)=4x4−4x2 by using the starting values x0=−2 and x0=−0.8, lying (−∞,−22).

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=−0.5 and x0=0.25, lying (−217,217).

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=0.8 and x0=2, lying (22,∞).

The zeros of the function f(x)=4x4−4x2 by using the starting values x0=−217 and x0=217.

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