Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/192

 Since $$f'(a) \ne 0$$, and $$f'(x)$$ is assumed as continuous, h may be chosen so small that $$f'(x)$$ will have the same sign as $$f'(a)$$ for all values of x in the interval [a - h, a + h]. Therefore $$f'(x_1)$$ has the same sign as $$f'(a)$$ (Chap. III). But x - a changes sign according as x is less or greater than a. Therefore, from (C), the difference

$$f(x) - f(a)$$

will also change sign, and, by (A) and (B), $$f(a)$$ will be neither a maximum nor a minimum. This result agrees with the discussion in § 82, where it was shown that for all values of x for which $$f(x)$$ is a maximum or a minimum, the first derivative $$f'(x)$$ must vanish.

II. Let $$f'(a) = 0$$, and $$f''(a) \ne 0$$.

From (C), &sect; 107, replacing b by x and transposing $$f(a)$$,

Since $$f(a) \ne 0$$, and $$f(x)$$ is assumed as continuous, we may choose our interval [a - h, a + h] so small that $$f(x_2)$$ will have the same sign as $$f(a)$$ (Chap. III). Also $$(x - a)^2$$ does not change sign. Therefore the second member of (D) will not change sign, and the difference

$$f(x) - f(a)$$

will have the same sign for all values of x in the interval [a - h, a + h], and, moreover, this sign will be the same as the sign of $$f''(a)$$. It therefore follows from our definitions (A) and (B) that

These conditions are the same as (21) and (22), §84.

III. Let $$f'(a) = f(a) = 0$$, and $$f'(a) \ne 0.$$

From (D), §107, replacing b by x and transposing $$f(a)$$,

As before, $$f(x_3)$$ will have the same sign as $$f(a)$$. But $$(x - a)^3$$ changes its sign from - to + as x increases through a. Therefore the difference

$$f(x) - f(a)$$

must change sign, and $$f(a)$$ is neither a maximum nor a minimum.