Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/269

In order to determine sufficient conditions that $f(a,b)$ shall be a maximum or a minimum, it is necessary to proceed to higher derivatives. To derive sufficient conditions for all cases is beyond the scope of this book.^[1] The following discussion, however, will suffice for all the problems given here.

Expanding $f(a+h,b+k)$ by Taylor's Theorem, (67), §148, replacing $x$ by $a$ and $y$ by $b$, we get

(D)

${\begin{aligned}f\left(a+h,b+k\right)=&f\left(a,b\right)+h{\frac {\partial f}{\partial x}}+k{\frac {\partial f}{\partial y}}\\&+{\frac {1}{2!}}\left(h^{2}{\frac {\partial ^{2}f}{\partial x^{2}}}+2hk{\frac {\partial ^{2}f}{\partial x\partial y}}+k^{2}{\frac {\partial ^{2}f}{\partial y^{2}}}\right)+R,\\\end{aligned}}$

where the partial derivatives are evaluated for $x=a,\quad y=b$, and $R$ denotes the sum of all the terms not written down. All such terms are of a degree higher than the second in $h$ and $k$.

Since ${\tfrac {\partial f}{\partial x}}=0$ and ${\tfrac {\partial f}{\partial y}}=0$, from (C), we get, after transposing $f(a,b)$,

(E)	$f\left(a+h,b+k\right)-f\left(a,b\right)={\frac {1}{2}}\left(h^{2}{\frac {\partial ^{2}f}{\partial x^{2}}}+2hk{\frac {\partial ^{2}f}{\partial x\partial y}}+k^{2}{\frac {\partial ^{2}f}{\partial y^{2}}}\right)+R.$

If $f(a,b)$ is a turning value, the expression on the left-hand side of must retain the same sign for all values of $h$ and $k$ sufficiently small in numerical value, $-$ the negative sign for a maximum value (see (A)) and the positive sign for a minimum value (see (B); i.e. $f(a,b)$ will be a maximum or a minimum according as the right-hand side of (E) is negative or positive. Now $R$ is of a degree higher than the second in $h$ and $k$. Hence as $h$ and $k$ diminish in numerical value, it seems plausible to conclude that the numerical value of $R$ will eventually become and remain less than the numerical value of the sum of the three terms of the second degree written down on the right-hand side (E).^[2] Then the sign of the right-hand side (and therefore also of the left-hand side) will be the same as the sign of the expression

(F)	$h^{2}{\frac {\partial f}{\partial x^{2}}}+2hk{\frac {\partial ^{2}f}{\partial x\partial y}}+k^{2}{\frac {\partial ^{2}f}{\partial y^{2}}}.$

But from Algebra we know that the quadratic expression

$h^{2}A+2hkC+k^{2}B$

always has the same sign as $A$ (or $B$) when $AB-C^{2}>0$.

1. See Cours d' Analyse, Vol. I by C. Jordan
2. Peano has shown that this conclusion does not always hold. See the article on "Maxima and Minima of Functions of Several Variables," by Professor James Pierpont in the Bulletin of the American Mathematical Society, Vol. IV.