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Chapter 4 | Applications of Derivatives
Sign of f ″ ( x ) =6 x −12 at Test Point
Interval
Test Point
Conclusion
(−∞, 2)
x =0
−
f is concave down
(2, ∞)
x =3
+
f is concave up.
We conclude that f is concave down over the interval (−∞, 2) and concave up over the interval (2, ∞). Since f changes concavity at x =2, the point ⎛ ⎝ 2, f (2) ⎞ ⎠ = (2, 32) is an inflection point. Figure 4.36 confirms the analytical results.
Figure 4.36 The given function has a point of inflection at (2, 32) where the graph changes concavity.
4.18
For f ( x ) =− x 3 + 3 2
x 2 +18 x , find all intervals where f is concave up and all intervals where f is
concave down.
We now summarize, in Table 4.1 , the information that the first and second derivatives of a function f provide about the graph of f , and illustrate this information in Figure 4.37 .
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