A GENERAL PRINCIPLE UNDERLYING EXTREME VALUE PROBLEMS - MAXIMA AND MINIMA - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER VII. MAXIMA AND MINIMA

*§2. A GENERAL PRINCIPLE UNDERLYING EXTREME VALUE PROBLEMS

1. The Principle

The preceding problems are examples of a general question which is best formulated in analytic language. If, in the problem of finding the extreme values of p + q, we denote by x, y the coordinates of the point R, by x1, y1 the coördinates of the fixed point P, and by x2, y2 those of Q, then

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and the problem is to find the extreme values of the function

f(x, y) = p + q.

This function is continuous everywhere in the plane, but the point with the coördinates x, y is restricted to the given curve C. This curve will be defined by an equation g(x, y) = 0; e.g. x2 + y2 – 1 = 0 if it is the unit circle. Our problem then is to find the extreme values of f(x, y) when x and yare restricted by the condition that g(x, y) = 0, and we shall consider this general type of problem.

To characterize the solutions, we consider the family of curves with the equations f(x, y) = c; that is, the curves given by equations of this form, where the constant c may have any value, the same for all points of any one curve of the family. Let us assume that one and only one curve of the family f(x, y) = c passes through each point of the plane, at least if we restrict ourselves to the vicinity of the curve C. Then as c changes, the curve f(x, y) = c will sweep out a part of the plane, and no point in this part will be touched twice in the sweeping process. (The curves x2y2 = c,x + y = c, and x = c are such families.) In particular, one curve of the family will pass through the point R1, where f(x, y) takes on its greatest value on C, and another one through the point R2 where f(x, y) takes on its least value. Let us call the greatest value a and the least value b. On one side of the curve f(x, y) = a the value of f(x, y) will be less than a. and on the other side greater than a. Since f(x, y) ≤ a on C, C must lie entirely on one side of the curve f(x, y) = a; hence it must be tangent to that curve at R1. Similarly, C must be tangent to the curve f(x, y) = b at R2. We thus have the general theorem: If at a point R on a curve C a function f(x, y) has an extreme value a, the curve f(x, y) = a is tangent to C at R.

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Fig. 187. Extrema of a function on a curve.

2. Examples

The results of the preceding section are easily seen to be special cases of this general theorem. If p + q is to have an extreme value, the function f(x, y) is p + q, and the curves f(x, y) = c are the confocal ellipses with foci P and Q. As predicted by the general theorem, the ellipses passing through the points on C where f(x, y) takes on its extreme values were seen to be tangent to C at these points. In the case where the extrema of pq are sought, the function f(x, y) is pq, the curves f(x, y) = c are the confocal hyperbolas with P and Q as their foci, and the hyperbolas passing through the points of extreme value of f(x, y) were seen to be tangent to C.

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Fig. 188. Confocal ellipses.

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Fig. 189. Confocal hyperbolas.

Another example is the following: Given a line segment PQ and a straight line L not intersecting the line segment. At what point of L will PQ subtend the greatest angle?

The function to be maximized here is the angle θ subtended by PQ from points on L. The angle subtended by PQ from any point R in the plane is a function θ = f(x, y) of the coördinates of R. From elementary geometry we know that the family of curves θ = f(x, y) = c is the family of circles through P and Q, since a chord of a circle subtends the same angle at all points of the circumference on the same side of the chord. As is seen from Figure 190, two of these circles will, in general, be tangent to L, with centers on opposite sides of PQ. One of the points of tangency gives the absolute maximum for θ, while the other point yields a “relative” maximum (that is, the value of θ will be less in a certain neighborhood of this point than at the point itself. The greater of the two maxima, the absolute maximum, is given by that point of tangency which lies in the acute angle formed by the extension of PQ and L, and the smaller one by the point which lies in the obtuse angle formed by these two lines. (The point where the extension of the segment PQ intersects L gives the minimum value of θ, zero.)

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Fig. 190. Point on L from which segment PQ appears largest.

As a generalization of this problem we may replace L by a curve C and seek the point R on C at which a given line segment PQ (not intersecting C) subtends the greatest or least angle. Here again, the circle through P, Q, and R must be tangent to C at R.