In the derivation of Newton’s method, to determine the formula for xi+1, the function f(x) is approximated using a first-order Taylor approximation centered at xi. This problem investigates what happens when you try to use a second-order Taylor approximation

Part A.

Let f(x) = 0;

suppose x= a+h

such that f(x) =f(a+h) = 0

By second order Taylor approximation, we get

f(a) + hf'±(a) + f''(a) = 0

±

So, we get the succeeding equation for Newton's method as

±

Part B.

It is evident that Newton's method fails in two cases, as:

1.  if f''(x) = 0

2. if f'(x)² is less than 2f(x)f''(x)    

Part C.

In case   is close to , the choice that shouldbe made instead of ± in part A is:

f'(x) =  ⇔

Part D.

As given = = h

or                 h = -

We get,

f(a) + hf'(a) +(h²/2)f''(a) = 0

or h² = -hf(a)/f'(a)

Also,             (-)² = -(-)(f()/f'())

So,                f(a) + hf'(a) - (f''(a)/2)(hf(a)/f'(a)) = 0

It becomes   h = -f(a)/f'(a) + (h/2)[f''(a)f(a)/(f(a))²]

Also,             = -f()/f'() + [( - )f''()f()]/[2(f'())²]

the answer is thin because within context, the stanza concludes the and frail tone the people have toward each other, and the fact that it would be thin, makes the most sense compared to "pliable" or "not hard or firm to the touch". hope this , good luck.

step-by-step explanation:

can i have brainliest