| δ2u | · | δ2u | > ( | δ2u | )2 |
| δx12 | δx22 | δx1δx2 |
for a maximum or minimum at all and δ2u/δx12 and δ2u/δx22 both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when δx1, δx2, ... δxn all vanish. If, in the case of two variables,
| δ2u | · | δ2u | = ( | δ2u | )2 |
| δx12 | δx22 | δx1δx2 |
then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of.
Take for instance the function u = x2 − xy2 + y2. Here the values x = 0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are (δx)2, and are never negative. Here δu = δx2 − δxδy2 + δy2, and by putting δx = 0 or an infinitesimal of the same order as δy2, we can make the sign of δu depend on that of δy2, and so be positive or negative as we please. On the other hand, if we take the function u = x2 − xy2 + y4, x = 0, y = 0 make zero a critical value of u, and here δu = δx2 − δxδy2 + δy4, which is always positive, because we can write it as the sum of two squares, viz. (δx − 1⁄2δy2)2 + 3⁄4δy4; so that in this case zero is a minimum value of u.
A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. If, for example, x is restricted to lie between the values a and b and φ′(x) = 0 has no roots in this interval, it follows that φ′(x) is one-signed as x increases from a to b, so that φ(x) is increasing or diminishing all the time, and the greatest and least values of φ(x) are φ(a) and φ(b), though neither of them is a critical value. Consider the following example: A person in a boat a miles from the nearest point of the beach wishes to reach as quickly as possible a point b miles from that point along the shore. The ratio of his rate of walking to his rate of rowing is cosec α. Where should he land?
Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be θ, so that OP = a secθ, and PB = b − a tan θ. If his rate of rowing is V miles an hour his time will be a sec θ/V + (b − a tan θ) sin α/V hours. Call this T. Then to the first power of δθ, δT = (a/V) sec2θ (sin θ − sin α)δθ, so that if AOB > α, δT and δθ have opposite signs from θ = 0 to θ = α, and the same signs from θ = α to θ = AOB. So that when AOB is > α, T decreases from θ = 0 to θ = α, and then increases, so that he should land at a point distant a tan α from A, unless a tan α > b. When this is the case, δT and δθ have opposite signs throughout the whole range of θ, so that T decreases as θ increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not.
The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.
In the case of a function of several variables, X1, x2, ... xn, not independent but connected by m functional relations u1 = 0, u2 = 0, ..., um = 0, we might proceed to eliminate m of the variables; but Lagrange’s “Method of undetermined Multipliers” is more elegant and generally more useful.
We have δu1 = 0, δu2 = 0, ..., δum = 0. Consider instead of δu, what is the same thing, viz., δu + λ1δu1 + λ2δu2 + ... + λmδum, where λ1, λ2, ... λm, are arbitrary multipliers. The terms of the first order in this expression are