Local Maxima and Minima

References are to Salas/Hille/Etgen’s Calculus, 8th Edition

We study the behavior of the scalar-valued function f(r) of the 2-dimensional vector variable r near a stationary point r_{0} (one where ∇f(r_{0}) = 0). We wish to determine whether f has a local maximum or minimum at r_{0}. We assume throughout that f has class C^{2} (all second-order partial derivatives of f exist and are continuous).

There is one situation that occurs naturally in n dimensions, for n ≥ 2.

Definition 1 The point r_{0} is a saddle point of f if it is stationary (∇f(r_{0}) = 0), but is neither a local maximum nor a local minimum of f. (For a good picture, see p. 916 or p. 834; you can fill in the horse.)

The second derivative We reduce to a one-dimensional problem by restricting attention to the values of f along a line through r_{0}. We choose any vector h = h_{1}i+h_{2}j and set g(t) = f(r_{0} + th). Then by the chain rule,

g′(t) = ∇f(r_{0} + th) • h = f_{x}(r_{0} + th) h_{1} + f_{y}(r_{0} + th) h_{2}.

In particular, g′(0) = 0, and Taylor’s Theorem simplifies to

g(t) = f(r_{0} + th) = g(0) + \frac{t^{2}}{2} g′′(θt) = f(r_{0}) + \frac{t^{2}}{2} g′′(θt),  (2)

for some θ (depending on t), where 0 < θ < 1. It is now clear by continuity that:

(a) If g′′(0) > 0, then g has a strict local minimum at t = 0. In other words, g(t) > g(0) for all small t ≠ 0. So g(t) does not have a local maximum at t = 0, and f(r) does not have a local maximum at r_{0}. (b) If g′′(0) < 0, then g has a strict local maximum at t = 0. In other words, g(t) < g(0) for all small t ≠ 0. So g(t) does not have a local minimum at t = 0, and f(r) does not have a local minimum at r_{0}. (c) If g′′(0) = 0, we have no information about g′′(θt) or g(t). (3)

We compute that

g′′(0) = Q(h) = Ah_{1}^{2} + 2Bh_{1}h_{2} + Ch_{2}^{2},  (4)

a quadratic form in h, which we abbreviate to Q(h), with coefficients

A = f_{xx}(r_{0}),  B = f_{xy}(r_{0}),  C = f_{yy}(r_{0}).

Classification There are six types of quadratic form Q(h):

(i) positive-definite: Q(h) > 0 for all h ≠ 0; (ii) positive-semidefinite: Q(h) takes both positive and zero values for h ≠ 0; (iii) negative-definite: Q(h) < 0 for all h ≠ 0; (iv) negative-semidefinite: Q(h) takes both negative and zero values for h ≠ 0;

(v) indefinite: Q(h) takes both positive and negative values (and hence also zero values) for h ≠ 0; (vi) zero: Q(h) = 0 for all h.

Except for (vi), the type of a given quadratic form Q(h) is not obvious in general.

Case 1: A ≠ 0. We complete the square, by rewriting equation (4) as

Q(h) = A (h_{1} + \frac{B}{A} h_{2})^{2} + (C − \frac{B^{2}}{A} ) h_{2}^{2} = A (h_{1} + \frac{B}{A} h_{2})^{2} − \frac{D}{A}h_{2}^{2},  (5)

where we introduce the discriminant D = B^{2} − AC of Q(h). [WARNING: Many books (with good reason) change the sign and consider AC−B^{2} rather than B^{2}−AC.] Since the expressions h_{2} and h_{1} + (B/A)h_{2} can be chosen independently, all we have to do is check the signs of the coefficients A and −D/A.

Case 2: A = 0. Here, D = B^{2}, and we write

Q(h) = (2Bh_{1} + Ch_{2})h_{2}.  (6)

Theorem 7 The behavior of f near r_{0} is given by the following table:

Type of Q(h)  Conditions  Local minimum? Local maximum?
Positive-definite  D < 0, A > 0  yes, strict  no
Positive-semidef. D = 0, (A > 0 or C > 0)  maybe  no
Negative-definite  D < 0, A < 0  no  yes, strict
Negative-semidef. D = 0, (A < 0 or C < 0)  no  maybe
Indefinite  D > 0  no, saddle point no, saddle point
Zero  A = B = C = 0  maybe  maybe

This includes everything in Theorem 15.5.3, and a little more. Simple examples show that all statements are best possible. These results remain valid in n dimensions (with the second column greatly generalized, in ways that are not obvious).

Proof We read off the conditions from equations (5) and (6). The “no” statements follow directly from (3) and equation (5) or (6). If there exists h such that Q(h) > 0, then f does not have a local maximum; if there exists h such that Q(h) < 0, then f does not have a local minimum.

For the positive-definite case, we use equation (2) to write

f(r_{0} + h) = g(1) = f(r_{0}) + \frac{1}{2}Q(h),

where Q(h) is given by equation (5), except that we must now evaluate A, B, C and D at r_{0} + θh rather than r_{0}. Since these functions are all continuous, we still have D < 0 and A > 0 for all small h, and it is clear that Q(h) > 0 for all small h ≠ 0.

The negative-definite case is entirely similar. Alternatively, we can simply consider −f instead of f.