CHAPTER XIV.

MAXIMA AND MINIMA-ONE INDEPENDENT VARIABLE.

343. Elementary Algebraical Methods.

Examples frequently occur in elementary algebra and geometry in which it is required to find whether any limitations exist to the admissible values of certain functions for real values of the variable or variables upon which they depend.

For example, the function x2-4x+9 may be written in the form (x-2)2+5,

from which it is at once apparent that the least admissible value of the expression is 5, the value which it assumes when x=2. For the square of a real quantity is essentially positive, and therefore any value of x other than 2 will give a greater value than 5 to the expression. considered.

As a second illustration let us investigate whether any limitation exists to the values of the expression

x2-x+1

x2+x+1

for real values of x.

Putting

we have

x2(1 − y) −x(1+y)+1−y=0,

an equation whose roots are real only when

i.e., when

(1+y)2>4(1—y)2,

(3y—1)(3—y) is positive;

i.e., when y lies between the values 3 and 3. It appears therefore that the given expression always lies in value between 3 and 1. Its maximum value is therefore 3 and its minimum 3.

344. Method of Projection.

Ex. Suppose it be required to determine geometrically the greatest triangle inscribed in a given ellipse.

It is obvious from elementary considerations that if the ellipse be projected orthogonally into a circle the greatest triangle inscribed in the given ellipse must project into the greatest triangle inscribed in a circle; and such a triangle is equilateral and the tangent to the circle at each angular point is parallel to the opposite side. This property of parallelism is a projective property, and therefore holds for the greatest triangle inscribed in the given ellipse.

Moreover

Area of greatest triangle inscribed in the ellipse

Area of ellipse

Area of equilateral triangle inscribed in a circle
Area of the circle

3/3

47

Hence the area of the greatest triangle inscribed in an

1

1. Show algebraically that the expression + cannot lie be

X

tween 2 and -2 for real values of x. Illustrate this geometrically

4. Show that the triangle of greatest area with given base and vertical angle is isosceles.

5. Show that the greatest chord passing through a point of intersection of two given circles is that which is drawn parallel to the line joining the centres.

6. If A, B be two given points on the same side of a given straight line and P be a point in the line, then AP+BP will be least when AP and BP are equally inclined to the straight line.

7. Show that the triangle of least perimeter inscribable in a given triangle is the pedal triangle.

8. If A, B, C be the angular points of a triangle and P any other point, then AP+BP+CP will be a minimum when each of the angles at P is 120°. [AP is a normal to the ellipse with foci B, C and passing through P.]

9. The diagonals of a maximum parallelogram inscribed in an ellipse are conjugate diameters of the ellipse.

10. If the sum of two varying positive quantities be constant show that their product is greatest when the quantities are equal. Extend this to the case of any number of positive quantities.

THE GENERAL PROBLEM.

345. Suppose x to be any independent variable capable of assuming any real value whatever, and let p(x) be any given function of x. Let the curve y = p(x) be represented in the adjoining figure, and let A, B, C, D, ... be those points on the curve at which the tangent is parallel to one of the co-ordinate axes.

Suppose an ordinate to travel from left to right along the axis of x. Then it will be seen that as the ordinate passes such points as A, C, or E it ceases to increase and begins to decrease; whilst when it passes through B, D, or Fit ceases to decrease and begins to increase. At each of the former set of points the ordinate is said to have a maximum value, whilst at the latter it is said to have a minimum value.

346. Points of Inflexion.

On inspection of Fig. 83 it will be at once obvious that at such points of inflexion as G or H, where the tangent

is parallel to one of the co-ordinate axes, there is neither a maximum nor a minimum ordinate. Near G, for instance, the ordinate increases up to a certain value NG, and then as it passes through G it continues to increase without any prior sensible decrease.

This point may however be considered as a combination of two such points as A and B in Fig. 82, the ordinate

increasing up to a certain value NG1, then decreasing through an indefinitely small and negligible interval to NG, and then increasing again as shown in the magnified figure (Fig. 84), the points G1, G, being ultimately coincident.

2

2 B

2