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), 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 &sect;81 and &sect;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 $$x = a, y = 5$$,

'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 &sect;81, &sect;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 $$x = a, \quad y = b$$.