[Put x=a+h, y=a+k, and expand in powers of h and k, and finally, after reduction, put h = 0, k = 0.]

32. Show that generally, if a function of two independent

0

variables take one of the singular forms etc., for certain

values of the variables, its value is truly indeterminate.

greater or less than m, u and b being both greater than unity.

49. Prove that if, when x is infinite, (x) = ∞, then will

52. Prove Lt-oh{aTM+a+hm+a+2h|m + + a + (n − 1)h TM" }

...

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

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 . It appears therefore that the given expression always lies in value between 3 and §. 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

Hence the area of the greatest triangle inscribed in an