Guide to Answer

 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 signs of h and k change unless (2) holds.

Suppose now that the condition (2) is satisfied at a certain point P. Then from Equation. (35) of the Section (Multivariate Calculus (part 1)) we have

sign[∆f(x0, y0)] = sign[h2fxx(x0, y0) + 2hkfxy(x0, y0) + k2fyy(x0, y0)] (3)

when h and k are sufficiently small, unless the bracketed quantity is zero. That quantity is a quadratic expression in h and k of the form Ah2 + 2Bhk + Ck2. When the discriminant B2 − AC is positive, and only in that case, there will be two distinct values of the ration h/k for which the expression is zero, the expression having one sign for intermediate values of h/k and the oppsite sign for all other values. Hence a necessary condition that f have either a relative maximum or a relative minimum at P (x0, y0) is that

δ=f f −f2 ≥0 at (x,y). (4) xxyy xy 00

If δ < 0 at a point P(x0,y0) where (2) is satisfied, then ∆f is positive for some h and k and negative for others. Such a point is called a saddle point. A typical sketch of a saddle is shown in the figure below.

If δ > 0 at (x0,y0), then clearly fxx and fyy must be either both positive or both negative at that point. Since ∆f(x0,y0) is of constant sign in either case, when h and k are sufficiently small, it follows from (3)) that the former case corresponds to a realative minimum ∆ > 0 and the latter