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 \(x = 0\).

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