Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/190

Equating (D) and (E), we get

${\begin{smallmatrix}{\frac {f(b)-f(a)}{b-a}}=f'(x_{1}),\end{smallmatrix}}$

which is the Theorem of Mean Value.

The student should draw curves (as the one in §105 to show that there may be more than one such point in the interval; and curves to illustrate, on the other hand, that the theorem may not be true if $f(x)$ becomes discontinuous for any value of x between a and b (Fig. a, §105), or if $f'(x)$ becomes discontinuous (Fig. b, §105).

Clearing (44) of fractions, we may also write the theorem in the form

(45)	$f(b)=f(a)+(b-a)f'(x_{1}).$
Let $b=a+\Delta a$; then $b-a=\Delta a$, and since $x_{1}$ is a number lying between a and b, we may write
	$x_{l}=a+\theta \cdot \Delta a,$
where $\theta$ is a positive proper fraction. Substituting in (45), we get another form of the Theorem of Mean Value.
(46)	${\begin{smallmatrix}f(a+\Delta a)-f(a)=\Delta af'(a+\theta \cdot \Delta a).0<\theta <1\end{smallmatrix}}$

107. The Extended Theorem of Mean Value.^[1] Following the method of the last section, let R be defined by the equation

(A)	${\begin{smallmatrix}f(b)-f(a)-(b-a)f'(a)-{\frac {1}{2}}(x-a)^{2}=0.\end{smallmatrix}}$
Let $F(x)$ be a function formed by replacing b by x in the left-hand member of (A); that is,
(B)	${\begin{smallmatrix}F(x)=f(x)-f(a)-(x-a)f'(a)-{\frac {1}{2}}(x-a)^{2}R.\end{smallmatrix}}$
From (A), $F(b)=0$; and from (B), $F(a)=0$; therefore, by Rolle's Theorem (§105), at least one value of x between a and b, say $x_{1}$ will cause $F'(X)$ to vanish. Hence, since
	$F'(x)=f'(x)-f'(a)-(x-a)R$, we get
	$F'(x_{1})=f'(x_{1})-f'(a)-(x_{1}-a)R=0.$
Since $F'(x_{1})=0$ and $F'(a)=0$, it is evident that $F'(x)$ also satisfies the conditions of Rolle's Theorem, so that its derivative, namely $F''(x)$, must vanish for at least one value of x between a and $x_{1}$, say $x_{2}$, and therefore $x_{2}$ also lies between a and b. But
	$F''(x)=f''(x)-R$; therefore $F''(x_{2})=f''(x_{2})-R=0$,
and	$R=f''(x_{2})$.

1. Also called the Extended Law of the Mean.