Chapter 4 | Applications of Derivatives
399
Sign of f ′
Sign of f ″
Is f increasing or decreasing? Concavity
Positive
Positive
Increasing
Concave up
Positive
Negative
Increasing
Concave down
Negative
Positive
Decreasing
Concave up
Negative
Negative
Decreasing
Concave down
Table 4.1 What Derivatives Tell Us about Graphs
Figure 4.37 Consider a twice-differentiable function f over an open interval I . If f ′( x ) >0 for all x ∈ I , the function is increasing over I . If f ′( x ) <0 for all x ∈ I , the function is decreasing over I . If f ″( x ) >0 for all x ∈ I , the function is concave up. If f ″( x ) <0 for all x ∈ I , the function is concave down on I . The Second Derivative Test The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative. We know that if a continuous function has a local extrema, it must occur at a critical point. However, a function need not have a local extrema at a critical point. Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. Let f be a twice-differentiable function such that f ′( a ) =0 and f ″ is continuous over an open interval I containing a . Suppose f ″( a ) <0. Since f ″ is continuous over I , f ″( x ) <0 for all x ∈ I ( Figure 4.38 ). Then, by Corollary 3, f ′ is a decreasing function over I . Since f ′( a ) =0, we conclude that for all x ∈ I , f ′( x ) >0 if x < a and f ′( x ) <0 if x > a . Therefore, by the first derivative test, f has a local maximum at x = a . On the other hand, suppose there exists a point b such that f ′( b ) =0 but f ″( b ) >0. Since f ″ is continuous over an open interval I containing b , then f ″( x ) >0 for all x ∈ I ( Figure 4.38 ). Then, by Corollary 3, f ′ is an increasing function over I . Since f ′( b ) =0, we conclude that for all x ∈ I , f ′( x ) <0 if x < b and f ′( x ) >0 if x > b . Therefore, by the first derivative test, f has a local minimum at x = b .
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