DEFINITION. A function f(x) is said to have a maximum value for x = a if f(a) is greater than all other values of f(x) when x is near x = a. The point on the graph whose coördinates are x = a and y = f(a) is called a maximum point. X If x increases through x = a, the value of f(x) will increase to the value ƒ(a), and then decrease. And the maximum point will be higher than the other points on the graph nearby. DEFINITION. A function f(x) is said to have a minimum value for x = a if f(a) is less than all other values of f(x) when x is near x = a. The point [a, f(a)] on the graph is called a minimum point. If x increases through x = a, the value of f(x) will decrease to the value f(a), and then increase. The minimum point will be lower than other points on the graph nearby. EXAMPLE 1. Discuss and plot the graph of the function 2x2 3x - 9. Let y = 2x2 - 3x - 9. To solve for x in terms of y we first write (1) in the form (1) (2) Symmetry. The tests for symmetry show that the graph is not symmetrical with respect to either axis or the origin. Values excluded. From (1), no values of x need be excluded, so that the graph extends indefinitely to the right and left. Equation (2) shows that we must exclude all values of y for which 81+8y <0. Hence we must exclude y <- 10%, so that no part of the curve lies more than 10 units below the x-axis. Intercepts. Setting y 0 in (2), the intercepts on the x-axis, or the zeros of the function, are found to be == Setting x = 0 in (1), the intercept on the y-axis is y 9. Asymptotes. There are no vertical or horizontal asymptotes. The variation of the function is readily discussed in connection with the graph. Zeros of the function. These are represented by the intercepts on the x-axis, OA and OB. The curve crosses the axis at A and B. Sign of the function. The graph is above the x-axis to the left of the point A and to the right of B. Hence 2x2 - 3x − 9 > 0 if x <- 1.5, or x > 3. The graph is below the x-axis between A and B, and hence = Maximum and minimum values. We saw above that all values of y less than 10 must be excluded. Hence y 10 is the smallest value of the functions. Substituting this value in (2), the corresponding value of x is seen to be. Hence the function has the minimum value - 101 when x = 3. This value of the function is represented by the ordinate ED, and the point D(, - 101) is a minimum point. 4 This function does not have a maximum value. Changes of the function. Since the graph falls to the right at the left of the point D, 2x2 - 3x-9 is decreasing as x increases if x < 3. And since the curve rises to the right of D, 2x2 - 3x - 9 is increasing as x increases if x > 3. The variation may also be stated as follows, but care must be exercised to see that none of the important elements are omitted. For a numerically large negative value of x the function is positive. As x increases to - 1.5, f(x) decreases to zero. It then becomes nega tive, and continues to decrease until it assumes its minimum value of 101 when x = 2. As x increases from to 3, the function is negative and increases to zero. As x increases beyond x = 3, f(x) is positive and increasing. 4 4 In this example, the function does not change sign unless increases through a zero of the function. But a function may change sign if x increases through a value for which the function 1 becomes infinite. Thus whose graph is given on page 27, X 4' changes sign, from negative to positive, if x increases through the value 4. The intercepts on the x-axis and the vertical asymptotes should therefore be determined before considering the sign of a function. In the example just cited, the vertical asymptote does not separate intervals in which the function increases or decreases, but it may do so. For example, the function 1/22 becomes infinite if x approaches zero, and hence the y-axis is a vertical asymptote. As x increases, this function increases if x is negative, and decreases if x is positive. This function also shows that a vertical asymptote need not separate intervals in which the function has opposite signs, for 1/x2 is positive for all real values of x. A good order for considering the elements of the variation of a function is as follows: Zeros of the function, Function becomes infinite, Sign of the function, Maxima and minima, Changes of the function. As the zeros and asymptotes are taken up in discussing the table of values, only the last three are new ideas. These properties of a function may be determined approximately by inspection of the graph of the function. This procedure is especially useful if the table of values or the graph be given and the functional relation itself is unknown. The accuracy of the results will depend upon the choice of units on the axes, the care with which the graph is drawn, and the closeness with which it is read. EXAMPLE 2. A thermograph is an instrument which records the temperature continuously by means of such a curve as in the figure. Discuss the variation of the temperature as a function of the time. Zeros of the function. The graph cuts the time axis at B and D, hence the temperature was zero at 3 A.M. and at 8:30 a.m. Sign of the function. The graph is above the time axis from A to B and from D to F and below from B to D. Hence the temperature was above zero from 12 Mid. to 3 A.M. and from 8.30 A.M. to 12 Mid. the next night, and below zero from 3 A.M. to 8:30 a.m. Maximum and minimum values. At C the ordinates cease to decrease and begin to increase. Hence, the temperature had a minimum value of about 14° at 6 A.M. At E the ordinates cease to increase and begin to decrease. Hence, the temperature had a maximum value of about +45° at 3 P.M. Changes of the function. The graph rises from C to E and falls from A to C and from E to F. Hence, the temperature increased from 6 A.M. to 3 P.M. and decreased from 12 Mid, to 6 A.M. and from 3 P.M. to 12 Mid. If it is not desired to treat each topic separately, the results might be stated as follows: The temperature at midnight was 10°. It decreased until it became zero at 3 A.M. and continued to decrease until 6 A.M., when it reached a minimum value of about 14°. It then increased, becoming zero at 8:30 A.M., until 3 P.M., when its maximum value was about 45°. From that time on it decreased to about 20° at midnight. The discussion of this function is continued in the next section. EXERCISES 1. Discuss the table of values, plot the graph, and determine the variation of each of the functions: 2. The surplus and shortage of railroad cars in thousands for the months of the years 1915 and 1916 were as follows. Plot the graph and discuss the function. 26, 38. 1915. 180, 279, 321, 327, 291, 299, 275, 265, 185, 78, 1916. 46, 21, — 20, 3, 33, 57, 52, 9, -19, -60, 9, -19, -60, -114, -107. 3. The following are the data for the surplus reserves of New York banks in millions for the months of the years indicated. Plot the graph and discuss the function. The Federal Reserve System was inaugurated November 16, 1914. 1914. 29, 32, 20, 17, 38, 40, 15, -29, -28, -0.4, 72, 117. 1915. 121, 135, 131, 156, 168, 185, 159, 177, 197, 179, 168, 155. 4. One side of a rectangle whose perimeter is 12 inches is x. Find the area as a function of x. Construct the graph of the function, discussing the table of values, and find the value of x if the area is a maximum. What is the maximum area? 5. A farmer wishes to fence off a poultry yard whose area is to be 6 square rods. If one dimension is x, express the perimeter (the amount of fencing needed) as a function of x. Discuss the table of values and plot the graph of the function. What will the dimensions be to require the least amount of fencing? How much should he purchase? 6. There are a number of diseases with continued fever in which the course of the temperature is sufficiently characteristic to furnish the diagnosis. Croupous pneumonia is one of these. (See Fig. 22.) Discuss the three functions. The zero for temperature would be the normal temperature 98°.4. By comparison of the three functions state in words some of the symptoms of the disease. 7. The data for the temperature curves of measles and scarlet fever |