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Unformatted text preview: Lecture 10  Concavity, The Second Derivative Test, and Opti mization Word Problems 10.1 Concavity and the SecondDerivative Test Intuition: a curve is concave up on an interval I if it looks like on I . It is concave down on I if it looks like . We need a more precise definition. So f is increasing on this interval. And f is decreasing on this interval. These two suggest the following precise definition of concavity. Definition 10.1 If f is a function, we say f is concave up on I if f is increasing on I . We say that f is concave down on I if f is decreasing on I . Theorem 10.1 Let f be a function. Then 1. If f 00 ( x ) > for all x in I , then f is concave up on I . 2. If f 00 ( x ) < for all x in I , then f is concave down on I . Note that the first graph above has a local min and the second has a local max. This leads to the following Theorem 10.2 (The SecondDerivative Test): Suppose that c is a critical point for the function f . Then 1. If f 00 ( c ) > , then f has a local min at c . 2. If f 00 ( c ) < , then f has a local max at c . 3. If f 00 ( c ) = 0 or f 00 ( c ) does not exist, then the test fails. Note that the second derivative test is easier to use, but sometimes fails. The first derivative test always works. Example: Let f ( x ) = 6 x 2 +6 . Find all intervals where f is concave up and all intervals where it is concave down. solution: We have to calculate the second derivative. f ( x ) = ( 6)( x 2 + 3) 2 (2 x ) = 12 x ( x 2 + 3) 2 f 00 ( x ) = 12( x 2 + 3) 2 ( 12 x )(2( x 2 + 3)2 x ) ( x 2 + 3) 4 = 12( x 2 + 3) + 48 x 2 ( x 2 + 3) 3 = 36 x 2 36 ( x 2 + 3) 3 = 36( x 1)( x + 1) ( x 2 + 3) 3 The denominator is never 0. f 00 is 0 at x = 1 , 1. Thus we can use test points as before to produce the following diagram: f 00 ( x ) + + f ( x ) 1 1 It turns out the graph looks like this Any point at which the concavity of a function changes from up to down or viceversa is...
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 Fall '09
 PIETERHOFSTRA
 Derivative

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