lecture10 - Lecture 10 - Concavity, The Second Derivative...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 10 - Concavity, The Second Derivative Test, and Opti- mization Word Problems 10.1 Concavity and the Second-Derivative 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 Second-Derivative 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 vice-versa is...
View Full Document

Page1 / 8

lecture10 - Lecture 10 - Concavity, The Second Derivative...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online