. Differential and integral calculus, an introductory course for colleges and engineering schools. at the right, while in the case of the minimum points, mand m,f(x) is â at the left and + at the right. Thus theorem 5is verified. If f(x) has the same sign on both sides of the pointat which it is 0 or oo, that point is a flex with flex-tangents parallelor perpendicular to OX. Such are the points 72 and I±. 78 DIFFERENTIAL CALCULUS 61 It is possible that f{x) should change sign by a spring insteadof by passing through 0 or oo. See Art. 50 (b). The point may still be a true maximum orminimum poi
. Differential and integral calculus, an introductory course for colleges and engineering schools. at the right, while in the case of the minimum points, mand m,f(x) is â at the left and + at the right. Thus theorem 5is verified. If f(x) has the same sign on both sides of the pointat which it is 0 or oo, that point is a flex with flex-tangents parallelor perpendicular to OX. Such are the points 72 and I±. 78 DIFFERENTIAL CALCULUS 61 It is possible that f{x) should change sign by a spring insteadof by passing through 0 or oo. See Art. 50 (b). The point may still be a true maximum orminimum point, as in the ac-companying figure. Curveshaving such discontinuitieswill not be met with in thisbook. Example 1. Let us find themaximum and minimum values of f(x) = 2x3-9x2+12x. Solution. f(x) = 6a:2 - 18 x+ 12 = Q(x -l)(x - 2). The roots of f(x) = 0 are 1 and 2. When x< 1, f(x) is +; when 1 < x 2,f(x) is +.Therefore by theorem 5,/(l) = 5 is a maximum, and /(2) = 4 is a 2. Let us find the maximum and minimum values of /(:r.)=(z2-l)i4 x. Solution. fix) = 3(^_i)i f(x) has a zero at x = 0, and infinities at x = ± these in the order of their magnitude, â 1, 0, 1, we see thatwhen x<â 1, f(x) is â ; when â 1 < x < 0, f(x) is +; then by theorem 5, f(x) is min. at x = â 1;when 0 < x 1, /(#) is +; then by theorem 5, f(x) is min. at x = 1. Therefore, the maximum value of /(x) is /(0) = 1, and the minimumvalues are/( â1) = 0 and/(l) = 0. 61. Exercises. Find the maximum and minimum values of the following func-tions : 1. /(£) = £3-3z+6. 3. y= z3+3x2+6z+2. 4. y = sin u = 1 5. y = 1 + z2 *2- 1 (x-a})(x-V)xx+1z2-4z+l* §62 MAXIMA AND MINIMA 79 9. Show that x(x2 â 5)* has one maximum and one Show that x(x2 - 7) 3 has two maxima and two minima. 62. Employment of the Second Derivative in DeterminingMaxima and Minima. Let Xi be a zero oif{x) = 0. If }{xx) isdecreasing , , fix) is 17 .~° \, by theorem 3, coro
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