Chapter 10.3, Problem 26E

To determine

To prove: The given statement for the condition.

Explanation of Solution

Given information:

Quadratic Approximations Just as we call the Taylor polynomial of order 1 generated by $f$ at $x=a$ the linearization of $f$ at $a$ , we call the Taylor polynomial of order 2 generated by $f$ at $x=a$ the quadratic approximation of $f$ at $a$ .

Calculation:

For some function $f(x)$ , if $f(x)$ has derivatives of all order in an open interval I. Then for $n=0,$ , 's Theorem with Remainder says, $f(x)=f(a)+R(x)$

Where, $a$ exists in interval $I$

  $R(x)=f^{\prime }(c)(x-a),$ , for some $c$ between $a$ and $x$ .

Thus,

  $f(x)=f(a)+f^{\prime }(c)(x-a)$ . Let $x=b,$ , which $f(b)=f(a)+f^{\prime }(c)(b-a)$

  $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$ For some c between a and b which is known as Mean value Theorem. Thus, Mean Value Theorem considered to be an special case of's Theorem.