In order that this expression may be positive, it is necessary to have 0.48 y 12.52. The two values of x which correspond respectively to the values y, and y, of y are [from (2)], 4 5 Y 2 nor Hence, as x increases from ∞ to +∞, y begins with the value 2, and returns to this same value, and is never greater than less than Yi If x = 4, the corresponding value of y is y = 7; if x = x,, if x = = 0, y = if x x = 5, y=1.06. Hence, the variation of the value of the fraction y as x increases from ∞ to +∞ will be represented in the following table: It may be observed that the denominator has two real roots, 1 = 2, x= 3, and therefore y becomes infinite for these roots. The fraction may be written in the form, If x increases from the value of y begins with 1 and returns to 1. For values of x very near 2, but less than 2, x2 is negative, x3 is negative, and 2+7 is positive, and therefore the fraction = 2. is positive; but for x very near 2, but greater than 2, x 2 is positive, 3 is negative, and +7 is positive, and therefore the frac.x — tion y changes sign from +∞ to -, as x passes through x = Similarly, as r, increasing in value, passes through 3, the fraction y changes from ∞ to +∞. = After clearing fractions and arranging the equation with respect tor, the result is Solve and get x= x(y-1)-5 yx+6y-7=0. = 5 y± √ (5 y) −4( y − 1)(6 y — 7) _ 5 y± √ y2+ 52 y − 28. 2 (y-1) In order that the values of x may be real, it is necessary to have where y1268/II and y=-26—81/II. Y 2 and for Hence, y can not take a value which lies between then (y-y) (y-1) would be negative and therefore z would be imaginary; but it can take any value less than y, or greater than y Therefore, Y1 is a minimum and a maximum value of y. = to x = 2, y A résumé of the preceding discussion gives us, as the variation of y, the following results: as increases from ∞ to x = x Y decreases from 1 to y1 0.533+; as x increases from x1 increases from 0.533 to +∞o. When x passes through the value = 2, y changes sign increases from x=2 to x=x2 y increases from ∞ to y = y 2 = -52.533; as x increases from x, to x = = 3, y decreases from y=y 2 to-∞o, and changes sign and becomes + ∞ as x passes through x=3; as x increases from x=3 to + ∞ y decreases from+to+1. The results of this discussion are exhibited in Fig. 16, in which the ordinate AP is the minimum value of y and the ordinate BQ is the negative maximum value of y. EXERCISE LXXX Trace the graphs of ཇ. 3. y = x2 − 5 x + 6. 2. y=-3x2 + 12 x 6. y=4x2+20 x 25. 4. y = √1 + x + √1 − x. Trace the graphs of the following, and mark in particular the points where the graph cuts the axes, and the maximum and minimum values of y. 12. 13. 14. 15. 16. 17. 18. 19. Show that the algebraically greatest and least values of (x2+2x-2) ÷ (x2 + 3 x + 5) Find the maximum and minimum values of y in examples 20-22: 23. 24. 25. Inscribe in a square the square of least area. Circumscribe about a square the square of greatest area. Inscribe a rectangle in a circle which has a given area and determine the greatest such rectangle. Ans. Square. 26. Find the sides of a right-angled triangle, given the perimeter and area. 27. Circumscribe about a circle the isosceles trapezium of mini mum area. 28. Find the sides of a right-angled triangle, given the hypotenuse and the sum of the legs. 29. Draw a tangent to a given circle which shall form with two given perpendicular tangents the triangle of minimum area. 30. A box is made from a rectangular piece of cardboard 11 inches by 15 inches by cutting out equal squares at the corners of the sheet, and then turning up the flaps. Show how to construct in this way the box of greatest capacity. 31. Find the volume of the greatest cylinder inscribed in a sphere of radius a. 32. Find the cylinder of least surface, the volume being constant. 33. Find the cylinder of maximum volume. the surface being given. BOOK V CHAPTER I RATIO AND PROPORTION 477. The ratio of one number to another is the quotient formed by dividing the first by the second. a Thus the ratio of a to b is ; and is also written a : b. 478. A ratio of equality is one whose terms are equal, as 4: 4. A ratio of greater inequality is one whose first term is greater than the second; as 7: 5. A ratio of less inequality is one whose first term is less than the second; as 4: 9. Inverse ratios are two ratios in which the first term of the one is the second term of the other, and vice versa; as 3:5 and 5: 3. The duplicate ratio of a given ratio is one whose terms are the squares of the terms of the given ratio. Thus a b2 is the duplicate ratio of a: b. The triplicate ratio of a given ratio is one whose terms are the cubes of the terms of the given ratio. Thus, a3 3 is the triplicate ratio of a b. The subduplicate ratio of a given ratio is one whose terms are the square roots of the terms of the given ratio. Thus, Va: Vb is the subduplicate ratio of a b. 479. Four quantities are said to be proportionals when the first is the same multiple, part, or parts, of the second, as the third is of the fourth; that is, if a = mb and c = md, or, what amounts to the same thing, |