Page 63 - u4
P. 63

P2: OSO/OVY
 P1: OSO/OVY
                                                             17:24
                            GO01962-Smith-v1.cls
 UAE-Math-Grade-12-Vol-1-SE-718383-ch4
         4-43          QC: OSO/OVY  T1: OSO   October 25, 2018  SECTION 4.5 • •  Concavity and the Second Derivative Test  253
              -    0    +             Solution Here, we have
                               4x
                   0                                        f (x) = 4x − 12x = 4x(x − 3)
                                                             ′
                                                                    3
                                                                                2
                -       0  +
                               ( x - √ 3)                               √      √
                       √ 3                                       = 4x(x −  3)(x +  3).
            -  0     +
                                                                              ′
                               ( x + √ 3)  We have drawn number lines for the factors of f (x) in the margin. From this, we
             - √ 3                    can see that
            -  0  +  0 -  0  +
                               f�(x)                               √       √
             - √ 3  0  √ 3                               { > 0, on (− 3, 0) ∪ ( 3, ∞)  f increasing.
                                                     ′
                                                     f (x)
                                                                       √
                -     0  +                                 < 0, on (−∞, − 3) ∪ (0,  √ 3).  f decreasing.
                             12(x - 1)
                      1                                   ′′       2
                                      Next, we have       f (x) = 12x − 12 = 12(x − 1)(x + 1).
            -   0     +
                             (x + 1)
               -1                     We have drawn number lines for the two factors in the margin. From this, we can
            +   0  -  0  -            see that
                             f�(x)
               -1     1                                   { > 0, on (−∞, −1) ∪ (1, ∞)  Concave up.
                                                      ′′
                                                     f (x)
          -  0  +   0  -   0  +                             < 0, on (−1, 1).       Concave down.
                                 f'(x)
           -√3      0     √3          For convenience, we have indicated the concavity below the bottom number line,
           +   0    -   0   +         with small concave up and concave down segments. Finally, observe that since the
                                 f''(x)
              -1        1             graph changes concavity at x =−1 and x = 1, there are inflection points located at
                                      (−1, −4) and (1, −4). Using all of this information, we are able to draw the graph
                     y                shown in Figure 4.57. For your convenience, we have reproduced the number lines
                                         ′
                                                 ′′
                                      for f (x) and f (x) together above the figure.
                   10
                                                                        ′′
                                         As we see in example 5.3, having f (x) = 0 does not imply the existence of an
                                      inflection point.
                   5
                                  x   EXAMPLE 5.3     A Graph with No Inflection Points
         -3                    3
                                                                   4
                                      Determine the concavity of f(x) = x and locate any inflection points.
                  -5
                                                                                               ′
                                                                                                      3
                                      Solution There’s nothing tricky about this function. We have f (x) = 4x and
                                                        ′
                                       ′′
                                                                             ′
                                               2
                 -10                  f (x) = 12x . Since f (x) > 0 for x > 0 and f (x) < 0 for x < 0, we know that f
                                                                                           ′′
                                      is increasing for x > 0 and decreasing for x < 0. Further, f (x) > 0 for all x ≠
                                              ′′
                 FIGURE 4.57          0, while f (0) = 0. So, the graph is concave up for x ≠ 0. Further, even though
                                       ′′
                y = x − 6x + 1        f (0) = 0, there is no inflection point at x = 0. We show a graph of the function
                         2
                    4
                                      in Figure 4.58.
                     y
                                         We now explore a connection between second derivatives and extrema. Suppose
                                          ′
                                      that f (c) = 0 and that the graph of f is concave down in some open interval containing
                                      c. Then, near x = c, the graph looks like that in Figure 4.59a and hence, f(c) is a local
                   4
                                                          ′
                                      maximum. Likewise, if f (c) = 0 and the graph of f is concave up in some open interval
                                      containing c, then near x = c, the graph looks like that in Figure 4.59b and hence, f(c)
                                      is a local minimum.
                   2
                                              y                                 y
                                  x                       f (c) = 0
         -2   -1          1    2                                                       f�(c) > 0
                 FIGURE 4.58
                    y = x 4
                                                                                                                    Copyright © McGraw-Hill Education
                                                         f  (c) < 0                        f�(c) = 0
                                                                    x                                 x
                                                        c                                 c
                                                 FIGURE 4.59a                      FIGURE 4.59b
                                                  Local maximum                     Local minimum
        272 | Lesson 4-5 | Concavity and the Second Derivative Test
   58   59   60   61   62   63   64   65   66   67   68