Chapter 4 | Applications of Derivatives
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these points, we divide the domain of f into smaller intervals and determine the sign of f ″ over each of these smaller intervals. If f ″ changes sign as we pass through a point x , then f changes concavity. It is important to remember that a function f may not change concavity at a point x even if f ″( x ) =0 or f ″( x ) is undefined. If, however, f does change concavity at a point a and f is continuous at a , we say the point ⎛ ⎝ a , f ( a ) ⎞ ⎠ is an inflection point of f .
Definition If f is continuous at a and f changes concavity at a , the point ⎛
⎝ a , f ( a ) ⎞
⎠ is an inflection point of f .
Figure 4.35 Since f ″( x ) >0 for x < a , the function f is concave up over the interval (−∞, a ). Since f ″( x ) <0 for x > a , the function f is concave down over the interval ( a , ∞). The point ⎛ ⎝ a , f ( a ) ⎞ ⎠ is an inflection point of f .
Example 4.19 Testing for Concavity
For the function f ( x ) = x 3 −6 x 2 +9 x +30, determine all intervals where f is concave up and all intervals where f is concave down. List all inflection points for f . Use a graphing utility to confirm your results. Solution To determine concavity, we need to find the second derivative f ″( x ). The first derivative is f ′( x ) =3 x 2 −12 x +9, so the second derivative is f ″( x ) =6 x −12. If the function changes concavity, it occurs either when f ″( x ) =0 or f ″( x ) is undefined. Since f ″ is defined for all real numbers x , we need only findwhere f ″( x ) =0. Solving the equation 6 x −12=0, we see that x =2 is the only place where f could change concavity. We now test points over the intervals (−∞, 2) and (2, ∞) to determine the concavity of f . The points x =0 and x =3 are test points for these intervals.
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