Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/270

 Applying this to (F), $$A=\tfrac{\partial^2 f}{\partial x^2},\quad B = \tfrac{\partial^2 f}{\partial y^2}, \quad C = \tfrac{\partial^2 f}{\partial x \partial y}$$ and we see that (F), and therefore also the left-hand member of (E), has the same sign as $$\tfrac{\partial^2 f}{\partial x^2}$$ ( or $$\tfrac{\partial^2 f}{\partial y^2}$$) when

$$\frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial y^2} - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2 > 0.$$

Hence the following rule for finding maximum and minimum values of a function $$f(x,y)$$.
 * . Solve the simultaneous equations

$$\frac{\partial f}{\partial x} = 0, \qquad \frac{\partial f}{\partial y} = 0.$$
 * . Calculate for these values of x and y the value of

$$\Delta = \frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial y^2} - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2.$$


 * The function will have a

The student should notice that this rule does not necessarily give all maximum and minimum values. For a pair of values of $$x$$ and $$y$$ determined by the First Step may cause $$\Delta$$ to vanish, and may lead to a maximum or a minimum or neither. Further investigation is therefore necessary for such values. The rule is, however, sufficient for solving many important examples. The question of maxima and minima of functions of three or more independent variables must be left to more advanced treatises. Examine the function $$3axy - x^3 - y^3$$ for maximum and minimum values.




 * Solution.
 * align="center"| $$ f(x,\ y) = 3 axy - x^3 -y^3.$$
 * First step.
 * align="center"| $$ \frac{\partial f}{\partial x} = 3ay - ex^2 = 0, \qquad \frac{\partial f}{\partial y} = 3ax - 3y^2 =0.$$
 * colspan="2" align="left"| Solving these two simultaneous equations, we get
 * colspan="2" align="center"|$$x = 0, \qquad x = a,$$
 * colspan="2" align="center"|$$y = 0; \qquad y = a.$$
 * }
 * colspan="2" align="center"|$$x = 0, \qquad x = a,$$
 * colspan="2" align="center"|$$y = 0; \qquad y = a.$$
 * }
 * }