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Chapter 4 | Applications of Derivatives
Figure 4.34 (a), (c) Since f ′ is increasing over the interval ( a , b ), we say f is concave up over ( a , b ). (b), (d) Since f ′ is decreasing over the interval ( a , b ), we say f is concave down over ( a , b ).
In general, without having the graph of a function f , how can we determine its concavity? By definition, a function f is concave up if f ′ is increasing. From Corollary 3, we know that if f ′ is a differentiable function, then f ′ is increasing if its derivative f ″( x ) >0. Therefore, a function f that is twice differentiable is concave up when f ″( x ) >0. Similarly, a function f is concave down if f ′ is decreasing. We know that a differentiable function f ′ is decreasing if its derivative f ″( x ) <0. Therefore, a twice-differentiable function f is concave down when f ″( x ) <0. Applying this logic is known as the concavity test .
Theorem 4.10: Test for Concavity Let f be a function that is twice differentiable over an interval I . i. If f ″( x ) >0 for all x ∈ I , then f is concave up over I . ii. If f ″( x ) <0 for all x ∈ I , then f is concave down over I .
We conclude that we can determine the concavity of a function f by looking at the second derivative of f . In addition, we observe that a function f can switch concavity ( Figure 4.35 ). However, a continuous function can switch concavity only at a point x if f ″( x ) =0 or f ″( x ) is undefined. Consequently, to determine the intervals where a function f is concave up and concave down, we look for those values of x where f ″( x ) =0 or f ″( x ) is undefined. When we have determined
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