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Unformatted text preview: wtm369 Homework 9 Helleloid (58250) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points The derivative of a function f is given for all x by f ( x ) = (3 x 2 + 3 x 36) parenleftBig 1 + g ( x ) 2 parenrightBig where g is some unspecified function. At which point(s) will f have a local maximum? 1. local maximum at x = 3 2. local maximum at x = 4 correct 3. local maximum at x = 4 , 3 4. local maximum at x = 4 5. local maximum at x = 3 Explanation: At a local maximum of f , the derivative f ( x ) will be zero, i.e. , 3( x 3)( x + 4) parenleftBig 1 + g ( x ) 2 parenrightBig = 0 . Thus the critical points of f occur only at x = 4 , 3. To classify these critical points we use the First Derivative test; this means looking at the sign of f ( x ). But we know that 1 + g ( x ) 2 is positive for all x , so we have only to look at the sign of the product 3( x 3)( x + 4) of the other two factors in f ( x ). Now the sign chart 4 3 + + for 3( x 3)( x + 4) shows that the graph of f is increasing on ( , 4), decreasing on ( 4 , 3), and increasing on (3 , ). Thus f has a local maximum at x = 4 . 002 10.0 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 negationslash = 0 ; C. has local maximum at x = 0 ; Which does f have? 1. All of them 2. B only 3. None of them 4. B and C only correct 5. C only 6. A and B only 7. A and C only 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 ( x ) = 2 9 x 4 / 3 . Consequently, A. not have: ( f ( x ) > , x negationslash = 0); wtm369 Homework 9 Helleloid (58250) 2 B. has: ( f ( x ) = (2 / 3) x 1 / 3 , x negationslash = 0); C. has: (see graph). 003 10.0 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 maximum? 1. x = b , c 2. x = c 3. x = b 4. none of a , b , c 5. x = a 6. x = c , a correct 7. x = a , b 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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- Fall '08