Khal

 1 Relative maxima, relative minima and saddle points The developments of the previous section (Multivariate Calculus (part 1)) are helpful in studying maxima and minima of functions of several variables. We restrict our attention here to functions f(x,y) of two variables. If we define ∆f (x, y) = f (x + h, y + k) − f (x, y) (1) we say that f has a relative minimum at P(x0,y0) if ∆f(x0,y0) ≥ 0 for all sufficiently small permissible h and k and that f has a relative maximum at P if instead ∆f(x0,y0) ≤ 0 for all such h and k. If the point P is an interior point of a region in which f, fx and fy exist, Equation (35) of the Section (Multivariate Calculus (part 1)) shows that a necessary condition that f assume a relative maximum or minimum at x0, y0 is that fx=fy=0 at (x0,y0). (2) For, when h and k are sufficiently small, the sign of ∆f(x0,y0) will be the same as the sign of hfx(x0, y0) + kfy(x0, y0) when this quantity is not zero, and clearly the sign of this quantity will change as the s