HW11.pdf - young(toy68 HW11 villafuerte altu(53785 This...

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young (toy68) – HW11 – villafuerte altu – (53785) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points 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 negationslash = 0 ; C. concave down on ( −∞ , 0) (0 , ) ; Which does f have? 1. A only 2. None of them 3. C only 4. All of them 5. A and B only 6. B only correct 7. B and C only 8. A and C 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: (see graph); B. has: ( f ( x ) = (2 / 3) x 1 / 3 , x negationslash = 0); C. not have: ( f ′′ ( x ) > 0 , x negationslash = 0). 002 10.0points Use the graph a b c of the derivative of f to locate the critical points x 0 at which f has neither a local maxi- mum nor a local minimum? 1. x 0 = c correct 2. x 0 = a, b 3. x 0 = b 4. x 0 = c, a 5. x 0 = b, c 6. x 0 = a, b, c 7. x 0 = a 8. none of 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 0 = a, b, and c . Now by the first derivative test, f will have
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young (toy68) – HW11 – villafuerte altu – (53785) 2 (i) a local maximum at x 0 if f ( x ) changes from positive to negative as x passes through x 0 ; (ii) a local minimum at x 0 if f ( x ) changes from negative to positive as x passes through x 0 ; (iii) neither a local maximum nor a local minimum at x 0 if f ( x ) does not change sign as x passes through x 0 . Consequently, by looking at the sign of f ( x ) near each of x 0 = a, b, and c we see that f has neither a local maximum nor a local minimum at x 0 = c . 003 10.0points If f is decreasing and its graph is concave down on (0 , 1), which of the following could be the graph of the derivative , f , of f ? 1. 1 f ( x ) 2. 1 f ( x ) 3. f ( x ) 1 cor- rect 4. f ( x ) 1 Explanation: The function f decreases when f < 0 on (0 , 1), and its graph is concave down when f ′′ < 0. Thus on (0 , 1) the graph of f lies below the x -axis and is decreasing. Of the four graphs, only 1 f ( x ) has these properties. 004 10.0points Ted makes a chart to help him analyze the continuous function y = f ( x ):
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young (toy68) – HW11 – villafuerte altu – (53785) 3 y y y ′′ x< 1 + x = 1 2 0 1 <x< 0 x = 0 1 1 0 <x< 2 + x = 2 2 DNE x> 2 + + Consider the following statements: A. f has a local minimum at x = 2. B. f has a local maximum at x = 1. C. f has a point of inflection at x = 0. Which are correct? 1. A and B only 2. all are true correct 3. C only 4. A and C only 5. B only 6. none are true 7. B and C only 8. A only Explanation: A. True: f is decreasing to left and increas- ing to right of x = 2. B. True: f increasing to left and decreasing to right of x = 1.
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