   Chapter 11.10, Problem 83E

Chapter
Section
Textbook Problem

Prove Taylor’s Inequality for n = 2 , that is, prove that if | f ″ ′ ( x ) | ≤ M for | x − a | ≤ d then | R 2 ( x ) | ≤ M 6 | x − a | 3    for  | x − a | ≥ d

To determine

To prove:

The Taylor’s inequality for n=2

Explanation

1) Concept:

If fn+1xM  for  x-ad, then the remainder Rnx of the Taylor series satisfies the inequality,

Rnx=fx-Tnx

Where Tnx is the polynomial of the degree n and Rnx is the remainder

2) Given:

3) Calculation:

Let

By simplifying we have,

f'''xM  for axa+d

Now integrating it

axf'''tdt axMdt

By using Part 2 of the Fundamental Theorem of Calculus

f''x-f''aMx-a

Integrate again from  a to x,

f''a is constant and can be taken out

Again by using Part 2 of the Fundamental Theorem of Calculus

f'x-f'a-f''ax-a12Mx-a2

Integrate again from  a to x

axf'tdt-axf

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