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order to use Theorem 23.1, we must assume that f is continuous at c and exists in an open interval around c. However, a more complicated argument can avoid that assumption.
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CHAP. 23]
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EXAMPLE Consider the function f (x) = 2x3 + x2 4x + 2. Then,
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f (x) = 6x 2 + 2x 4 = 2(3x 2 + x 2) = 2(3x 2)(x + 1) Hence, if f (x) = 0, then 3x 2 = 0 or x + 1 = 0; that is, x = 2 or x = 1. Now f (x) = 12x + 2. Hence, 3 f ( 1) = 12( 1) + 2 = 12 + 2 = 10 < 0 f
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= 12( 2 ) + 2 = 8 + 2 = 10 > 0 3
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Since f ( 1) < 0, f has a relative maximum at x = 1, with f ( 1) = 2( 1)3 + ( 1)2 4( 1) + 2 = 2 + 1 + 4 + 2 = 5 Since f
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> 0, f has a relative minimum at x = 2 , with 3 f 2 3 =2 2 3 + 3 2 2 2 4 3 3 +2= 16 4 8 16 12 72 54 10 + +2= + = 27 9 3 27 27 27 27 27
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The graph of f is shown in Fig. 23-6. Now because f (x) = 12x + 2 = 12 x + 1 6 = 12 x 1 6
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f (x) > 0 when x > 1 , and f (x) < 0 when x < 1 . Hence, the curve is concave upward for x > 1 and concave downward 6 6 6 for x < 1 . So there must be an in ection point I, where x = 1 . 6 6
Fig. 23-6
From Problem 9.1 we know that
x +
lim f (x) =
x +
lim 2x 3 = +
x
lim f (x) =
x
lim 2x 3 =
Thus, the curve moves upward without bound toward the right, and downward without bound toward the left.
APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
[CHAP. 23
The second-derivative test tells us nothing when f (c) = 0 and f (c) = 0. This is shown by the examples in Fig. 23-7, where, in each case, f (0) = f (0) = 0.
Fig. 23-7 To distinguish among the four cases shown in Fig. 23-7, consider the sign of the derivative f just to the left and just to the right of the critical point. Recalling that the sign of the derivative is the sign of the slope of the tangent line, we have the four combinations shown in Fig. 23-8. These lead directly to Theorem 23-4. Theorem 23.4 (First-Derivative Test for Relative Extrema): Assume f (c) = 0. { , +} If f is negative to the left of c and positive to the right of c, then f has a relative minimum at c. {+, } If f is positive to the left of c and negative to the right of c, then f has a relative maximum at c. {+, +} and { , } If f has the same signs to the left and right of c, then f has an in ection point at c. 23.3 GRAPH SKETCHING
We are now equipped to sketch the graphs of a great variety of functions. The most important features of such graphs are: (i) Relative extrema (if any) (ii) In ection points (if any)
CHAP. 23]
APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
Fig. 23-8 (iii) Concavity (iv) Vertical and horizontal asymptotes (if any) (v) Behavior as x approaches + and The procedure was illustrated for the function f (x) = 2x 3 + x 2 4x + 2 in Section 23.2. An additional example follows. EXAMPLE Sketch the graph of the rational function
x f (x) = 2 x +1 First of all, the function is odd [that is, f ( x) = f (x)], so that it need be graphed only for positive x. The graph is then completed by re ection in the origin (see Section 7.3). Compute the rst two derivatives of f , f (x) = (x 2 + 1)Dx (x) xDx (x 2 + 1) x 2 + 1 x(2x) 1 x2 = = 2 2 + 1)2 2 + 1)2 (x (x (x + 1)2 (x 2 + 1)2 Dx (1 x 2 ) (1 x 2 )Dx ((x 2 + 1)2 ) (x 2 + 1)4
f (x) = Dx f (x) = =
2x(x 3)(x + 2x(x 2 + 1)[x 2 + 1 + 2(1 x 2 )] 2x(3 x 2 ) = = = (x 2 + 1)4 (x 2 + 1)3 (x 2 + 1)3
(x 2 + 1)2 ( 2x) (1 x 2 )[2(x 2 + 1)(2x)] (x 2 + 1)4
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