Chapter 4 | Applications of Derivatives
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of these points. For example, let’s choose x =−2, x =0, and x =4 as test points. Interval Test Point Sign of f ′ ( x ) =3 ( x −3 )( x +1 ) at Test Point
Conclusion
(−∞, −1)
(+)(−)(−) = +
x =−2
f is increasing.
(−1, 3)
(+)(−)(+) = −
x =0
f is decreasing.
(3, ∞)
(+)(+)(+) = +
x =4
f is increasing.
Step 3. Since f ′ switches sign from positive to negative as x increases through 1, f has a local maximum at x =−1. Since f ′ switches sign from negative to positive as x increases through 3, f has a local minimum at x =3. These analytical results agree with the following graph.
Figure 4.32 The function f has a maximum at x =−1 and a minimum at x =3
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