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Chapter 4 | Applications of Derivatives
4.5 | Derivatives and the Shape of a Graph Learning Objectives 4.5.1 Explain how the sign of the first derivative affects the shape of a function’s graph. 4.5.2 State the first derivative test for critical points. 4.5.3 Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. 4.5.4 Explain the concavity test for a function over an open interval. 4.5.5 Explain the relationship between a function and its first and second derivatives. 4.5.6 State the second derivative test for local extrema. Earlier in this chapter we stated that if a function f has a local extremum at a point c , then c must be a critical point of f . However, a function is not guaranteed to have a local extremum at a critical point. For example, f ( x ) = x 3 has a critical point at x =0 since f ′( x ) =3 x 2 is zero at x =0, but f does not have a local extremum at x =0. Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward. The First Derivative Test Corollary 3 of the Mean Value Theorem showed that if the derivative of a function is positive over an interval I then the function is increasing over I . On the other hand, if the derivative of the function is negative over an interval I , then the function is decreasing over I as shown in the following figure.
Figure 4.30 Both functions are increasing over the interval ( a , b ). At each point x , the derivative f ′( x ) >0. Both functions are decreasing over the interval ( a , b ). At each point x , the derivative f ′( x ) <0.
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