When determining critical numbers and/or stationary numbers you need to show each of the following to earn full credit.

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1 Definition Critical Numbers/Stationary Numbers A critical number of f, 0 x, is a number in the domain of f where either f x 0 0 or undefined. If f x 0 0, then the number x 0 is also called a stationary number of f. f x 0 is When determining critical numbers and/or stationary numbers you need to show each of the following to earn full credit. 1. The domain of the given function. 2. A fully factored version of the first derivative; remember if there are two or more terms in the first derivative and one or more terms is a fraction, you need to combine all of the terms after building common denominators between the terms. Your simplified expression must also contain no negative exponents. 3. A statement asserting where, over the domain of the given function, the first derivative is equal to 0. If finding critical numbers, you also need to state where, over the domain of the given function, the first derivative is undefined. Example 4. A complete 1 (and appropriate) conclusion. Find the critical numbers of the function g t 2 3t 1 t 1 2. Derivative Analysis 1

2 Example 2 Let g t t 7 3 t 8. Answer each of the following questions. Use your calculator to take all derivatives and to perform, all algebra. Show all relevant work in an organized and well documented manner. a. What are the critical numbers of g? b. State all local maximum and minimum points of g after first performing a first derivative test. 2 D erivative Analysis

3 c. State all inflection points of g after first performing a concavity test. Derivative Analysis 3

4 Example 3 ln x 5 Suppose that gx. Find the stationary numbers of g and then perform a second x 5 derivative test at each stationary point. Work this problem without your calculator. Theorem The Second Derivative Test a If f 0 and 0 If f a 0 and a 0 a If f 0 and 0 f a, then f has a local maximum point at x a. f, then f has a local minimum point at x a. f a, then this test is inconclusive. 4 D erivative Analysis

5 Example 4: Complete Table 1 Table 1: Inconclusive Second Derivative Tests f x and f f x x 4 6 f xx 4 5 f x x f x and f 4 Sign test on f x (informal documentation) Graphical behavior of y f x at x 4. Sooo... what conclusion can we make about a function at a point where both the first and second derivative of the function evaluate to 0? Derivative Analysis 5

6 Example t 2t 13t Find, if they exist, the absolute maximum and minimum values of g t 10t over the interval 0,10. No calculator usage other than for arithmetic purposes. The Absolute Max/Min Theorem If f is continuous over ab,, then f must have an absolute maximum value and an absolute minimum value on ab,. Furthermore, these values must occur either at critical points on ab, or at a or at b. 6 D erivative Analysis

7 Example 6 Use appropriate derivative based techniques to determine, if they exist, the absolute maximum and minimum values of f x x 2/ x 1 graphing feature of your calculator. 20 over the interval 1, 29. Check your result with the Derivative Analysis 7

8 Example7 A farmer wishes to construct a rectangular pen with one side of the pen along a perfectly straight river bank; no fencing is required along the side of the pen that abuts the river. The farmer has 320 feet of fencing with which he can enclose the pen. Find the dimensions of the pen which should be constructed so that the pen has maximal area. 8 D erivative Analysis