Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/191

Substituting this result in (A), we get

(C)

${\begin{smallmatrix}f(b)=f(a)+(b-a)f'(a)+{\frac {1}{2!}}(b-a)^{2}f''(x_{2}).\end{smallmatrix}}$

$a<x_{2}$

In the same manner, if we define S by means of the equation

${\begin{smallmatrix}f(b)\ -\ f(a)\ -\ (b-a)f'(a)\ -\ {\frac {1}{2!}}(b-a)^{2}f''(a)\ -\ {\frac {1}{3!}}(b-a)^{2}f''(a)S\ =\ 0,\end{smallmatrix}}$

we can derive the equation

(D)	$f(b)\ =\ f(a)$	$+(b-a)f'(a)+{\frac {1}{2!}}(b-a)^{2}f''(a)$
		$+{\frac {1}{3!}}(b-a)^{3}f'''(x_{3}),$	$a\ <x_{3}<\ b$

where $x_{3}$ lies between a and b.

By continuing this process we get the general result,

(E)	$f(b)\ =\ f(a)$	$+{\frac {(b-a)}{1!}}f'(a)+{\frac {(b-a)^{2}}{2!}}f''(a)$
		$+{\frac {(b-a)^{3}}{3!}}f'''(a)+\cdots +{\frac {(b-a)^{(n-1)}}{(n-1)!}}f^{(n-1)}(a)$
		$+{\frac {(b-a)^{n}}{n!}}f^{(n)}(x_{1}),$	$a\ <x_{1}<\ b$

where $x_{1}$ lies between a and b. (E) is called the Extended Theorem of Mean Value.

108. Maxima and minima treated analytically. By making use of the results of the last two sections we can now give a general discussion of maxima and minima of functions of a single independent variable.

Given the function $f(x)$. Let h be a positive number as small as we please; then the definitions given in § 82, may be stated as follows:

If, for all values of x different from a in the interval [a - h, a + h],

(A)	$f(x)-f(a)=$ a negative number,

then $f(x)$ is said to be a maximum when $x=a$.

If, on the other hand,

(B)	$f(x)-f(a)=$ a positive number,

then $f(x)$ is said to be a minimum when x = a.

Consider the following cases:

I. Let $f'(a)\neq 0.$

From (45), §106, replacing b by x and transposing $f(a)$,

(C)

$f(x)-f(a)=(x-a)f'(x_{1}).$

$a<x_{1}<x$