A. Maxima and minima. Taylor’s Theorem may be applied to give greater theoretical completeness to the tests of Ch. VI, §§ 122-123, though the results are not of much practical importance. It will be remembered that, assuming that $$\phi(x)$$ has derivatives of the first two orders, we stated the following as being sufficient conditions for a maximum or minimum of $$\phi(x)$$ at $$x = \xi$$: for a maximum, $$\phi'(\xi) = 0$$, $$\phi”(\xi) < 0$$; for a minimum, $$\phi'(\xi) = 0$$, $$\phi”(\xi) > 0$$. It is evident that these tests fail if $$\phi”(\xi)$$ as well as $$\phi'(\xi)$$ is zero.

Let us suppose that the first $$n$$ derivatives $\phi'(x),\quad \phi”(x),\ \dots,\quad \phi^{(n)}(x)$ are continuous, and that all save the last vanish when $$x = \xi$$. Then, for sufficiently small values of $$h$$, $\phi(\xi + h) – \phi(\xi) = \frac{h^{n}}{n!} \phi^{(n)} (\xi + \theta_{n} h).$ In order that there should be a maximum or a minimum this expression must be of constant sign for all sufficiently small values of $$h$$, positive or negative. This evidently requires that $$n$$ should be even. And if $$n$$ is even there will be a maximum or a minimum according as $$\phi^{(n)}(\xi)$$ is negative or positive.

Thus we obtain the test: if there is to be a maximum or minimum the first derivative which does not vanish must be an even derivative, and there will be a maximum if it is negative, a minimum if it is positive.

Example LVII

1. Verify the result when $$\phi(x) = (x – a)^{m}$$, $$m$$ being a positive integer, and $$\xi = a$$.

2. Test the function $$(x – a)^{m} (x – b)^{n}$$, where $$m$$ and $$n$$ are positive integers, for maxima and minima at the points $$x = a$$, $$x = b$$. Draw graphs of the different possible forms of the curve $$y = (x – a)^{m} (x – b)^{n}$$.

3. Test the functions $$\sin x – x$$, $$\sin x – x + \dfrac{x^{3}}{6}$$, $$\sin x – x + \dfrac{x^{3}}{6} – \dfrac{x^{5}}{120}$$, …, $$\cos x – 1$$, $$\cos x – 1 + \dfrac{x^{2}}{2}$$, $$\cos x – 1 + \dfrac{x^{2}}{2} – \dfrac{x^{4}}{24}$$, … for maxima or minima at $$x = 0$$.