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Unformatted text preview: Jolley, Garrett Homework 10 Due: Oct 30 2007, 3:00 am Inst: R Heitmann 1 This printout should have 21 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points The derivative of a function f is given by f ( x ) = ( x 2 5 x 14) g ( x ) for some unspecified function g such that g ( x ) > 0 for all x . At which point(s) does f have a local minimum? 1. local minimum at x = 2 2. local minimum at x = 7 correct 3. local minimum at x = 2 4. local minimum at x = 7 5. local minimum at x = 2 , 7 Explanation: At a local minimum of f , the derivative f ( x ) will be zero, i.e. , ( x 7)( x + 2) g ( x ) = 0 . Thus the critical points of f occur only at x = 2 , 7. To classify these critical points we use the First Derivative test; this means looking at the sign of f ( x ). But we know that g ( x ) > 0 for all x , so we have only to look at the sign of the product ( x 7)( x + 2) of the other two factors in f ( x ). Now the sign chart 2 7 + + for ( x 7)( x + 2) shows that the graph of f is increasing on ( , 2), decreasing on ( 2 , 7), and increasing on (7 , ). Conse quently, f has a local minimum at x = 7 , . keywords: local minimum, First Derivative Test critical points, sign chart, conceptual, 002 (part 1 of 1) 10 points Let f be the function defined by f ( x ) = 5 x 2 / 3 . Consider the following properties: A. concave down on ( , 0) (0 , ); B. derivative exists for all x ; C. has local minimum at x = 0; Which does f have? 1. A and B only 2. All of them 3. A and C only 4. B and C only 5. A only 6. None of them correct 7. C only 8. B only Explanation: The graph of f is 2 4 2 4 2 4 On the other hand, after differentiation, f ( x ) = 2 3 x 1 / 3 , f 00 ( x ) = 2 9 x 4 / 3 . Jolley, Garrett Homework 10 Due: Oct 30 2007, 3:00 am Inst: R Heitmann 2 Consequently, A. not have: ( f 00 ( x ) > , x 6 = 0); B. not have: ( f ( x ) = (2 / 3) x 1 / 3 , x 6 = 0; C. not have: (see graph). keywords: concavity, local maximum, True/False, graph 003 (part 1 of 1) 10 points Use the graph a b c of the derivative of f to locate the critical points x at which f does not have a local minimum? 1. x = b, c correct 2. x = c, a 3. x = c 4. x = a 5. none of a, b, c 6. x = a, b 7. x = a, b, c 8. x = b Explanation: Since the graph of f ( x ) has no holes, the only critical points of f occur at the x intercepts of the graph of f , i.e. , at x = a, b, and c . Now by the first derivative test, f will have (i) a local maximum at x if f ( x ) changes from positive to negative as x passes through x ; (ii) a local minimum at x if f ( x ) changes from negative to positive as x passes through x ; (iii) neither a local maximum nor a local minimum at x if f ( x ) does not change sign as x passes through x ....
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This note was uploaded on 10/29/2008 for the course M 408k taught by Professor Schultz during the Spring '08 term at University of Texas at Austin.
 Spring '08
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