Chapter 4 | Applications of Derivatives
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Figure 4.33 The function f has a local minimum at x =−1 and a local maximum at x =1.
Use the first derivative test to find all local extrema for f ( x ) = x −1 3 .
4.17
Concavity and Points of Inflection We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function. Figure 4.34 (a) shows a function f with a graph that curves upward. As x increases, the slope of the tangent line increases. Thus, since the derivative increases as x increases, f ′ is an increasing function. We say this function f is concave up. Figure 4.34 (b) shows a function f that curves downward. As x increases, the slope of the tangent line decreases. Since the derivative decreases as x increases, f ′ is a decreasing function. We say this function f is concave down. Definition Let f be a function that is differentiable over an open interval I . If f ′ is increasing over I , we say f is concave up over I . If f ′ is decreasing over I , we say f is concave down over I .
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