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

 Substituting this result in (A), we get

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

we can derive the equation

where $$x_3$$ lies between a and b.

By continuing this process we get the general result,

where $$x_1$$ lies between a and b. (E) is called the Extended Theorem of Mean Value.

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],

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

If, on the other hand,

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

Consider the following cases:

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

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