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Chapter 4 | Applications of Derivatives
Note this theorem does not claim that a function f must have a local extremum at a critical point. Rather, it states that critical points are candidates for local extrema. For example, consider the function f ( x ) = x 3 . We have f ′( x ) =3 x 2 =0 when x =0. Therefore, x =0 is a critical point. However, f ( x ) = x 3 is increasing over (−∞, ∞), and thus f does not have a local extremum at x =0. In Figure 4.15 , we see several different possibilities for critical points. In some of these cases, the functions have local extrema at critical points, whereas in other cases the functions do not. Note that these graphs do not show all possibilities for the behavior of a function at a critical point.
Figure 4.15 (a–e) A function f has a critical point at c if f ′( c ) =0 or f ′( c ) is undefined. A function may or may not have a local extremum at a critical point.
Later in this chapter we look at analytical methods for determining whether a function actually has a local extremum at a critical point. For now, let’s turn our attention to finding critical points. We will use graphical observations to determine whether a critical point is associated with a local extremum. Example 4.12 Locating Critical Points For each of the following functions, find all critical points. Use a graphing utility to determine whether the function has a local extremum at each of the critical points. a. f ( x ) = 1 3 x 3 − 5 2 x 2 +4 x b. f ( x ) = ⎛ ⎝ x 2 −1 ⎞ ⎠ 3
c. f ( x ) = 4 x
1+ x 2
Solution
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