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Unformatted text preview: Tran, Tony – Homework 10 – Due: Oct 30 2007, 3:00 am – Inst: Samuels 1 This print-out should have 21 questions. Multiple-choice 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- 6) 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 = 6 2. local minimum at x = 1 correct 3. local minimum at x =- 1 4. local minimum at x =- 6 5. local minimum at x =- 6 , 1 Explanation: At a local minimum of f , the derivative f ( x ) will be zero, i.e. , ( x- 1)( x + 6) g ( x ) = 0 . Thus the critical points of f occur only at x =- 6 , 1. 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- 1)( x + 6) of the other two factors in f ( x ). Now the sign chart- 6 1 + +- for ( x- 1)( x + 6) shows that the graph of f is increasing on (-∞ ,- 6), decreasing on (- 6 , 1), and increasing on (1 , ∞ ). Conse- quently, f has a local minimum at x = 1 , . 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. has local minimum at x = 0; B. derivative exists for all x ; C. concave up on (-∞ , 0) ∪ (0 , ∞ ); Which does f have? 1. B only 2. All of them 3. A and C only 4. C only correct 5. B and C only 6. A and B only 7. None of them 8. A 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 . Tran, Tony – Homework 10 – Due: Oct 30 2007, 3:00 am – Inst: Samuels 2 Consequently, A. not have: (see graph); B. not have: ( f ( x ) =- (2 / 3) x- 1 / 3 , x 6 = 0; C. has: ( f 00 ( x ) > , x 6 = 0). 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 has either a local maxi- mum or a local minimum? 1. x = b, c 2. x = a 3. x = c, a 4. x = b 5. none of a, b, c 6. x = c 7. x = a, b correct 8. x = a, b, c 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 04/27/2008 for the course M 408k taught by Professor Schultz during the Fall '08 term at University of Texas.
- Fall '08