Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/268

is a maximum point when it is "higher" than all other points on the surface in its neighborhood, the coordinate plane $XOY$ being assumed horizontal. Similarly, $P'$ is a minimum point on the surface when it is "lower" than all other points on the surface in its neighborhood. It is therefore evident that all vertical planes through P cut the surface in curves (as $APE$ or $DPE$ in the figure),

Wag-139-Maxima and Minima of Two Functions.png

each of which has a maximum ordinate $z(=MP)$ at $P$. In the same manner all vertical planes through $P'$ cut the surface in curves (as $BP'C$ or $FP'G$), each of which has a minimum ordinate $z(=NP')$ at $P'$. Also, any contour (as $HIJK$) cut out of the surface by a horizontal plane in the immediate neighborhood of $P$ must be a small closed curve. Similarly, we have the contour $LSRT$ near the minimum point $P'$.

It was shown in §81 and §82, that a necessary condition that a function of one variable should have a maximum or a minimum for a given value of the variable was that its first derivative should be zero for the given value of the variable. Similarly, for a function $f(x,y)$ of two independent variables, a necessary condition that $f(a,b)$ should be a maximum or a minimum (i.e. a turning value) is that for $x=a,y=5$,

(C)	${\frac {\partial f}{\partial x}}=0,\qquad {\frac {\partial f}{\partial y}}=0.$

'Proof. Evidently (A) and (B) must hold when $k=0$; that is,

$f(a+h,\ b)-f(a,\ b)$

is always negative or always positive for all values of $h$ sufficiently small numerically. By §81, §82, a necessary condition for this is that ${\tfrac {d}{dx}}f(x,b)$ shall vanish for $x=a$, or, what amounts to the same thing, ${\tfrac {\partial }{\partial x}}f(x,y)$ shall vanish for $x=a,\quad y=b$. Similarly, (A) and (B) must hold when $h=0$, giving as a second necessary condition that ${\tfrac {\partial }{\partial y}}f(x,y)$ shall vanish for $x=a,\quad y=b$.