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lecture25

# lecture25 - Math 006(Lecture 25 Second Derivative Denition...

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Math 006 (Lecture 25) Second Derivative Definition 1. For y = f ( x ), the second derivative of f , is d 2 f dx 2 ( x ) = f ( x ) = d dx f ( x ) . Example 1. Find the second derivatives of (a) f ( x ) = 3 x 2 + 3 x 3 + 1 x 7 (b) y = x 3 + x (c) y = x + 1 x 2 + 1 1

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Concavity Example 2. Find the first and the second derivatives of f ( x ). (a) f ( x ) = x 2 (b) f ( x ) = x Definition 2. Concavity of f ( x ) 1. The graph of a function is concave upward on the interval a < x < b if f ( x ) is increasing on a < x < b . 2. The graph of a function is concave downward on the interval a < x < b if f ( x ) is decreasing on a < x < b . Theorem 1. For the interval a < x < b , 1. f ( x ) is an concave upward on a < x < b if and only if f ( x ) > 0 on a < x < b . 2. f ( x ) is an concave downward on a < x < b if and only if f ( x ) < 0 on a < x < b . Example 3. Determine the interval(s) where the graph of
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lecture25 - Math 006(Lecture 25 Second Derivative Denition...

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