Section 4.3-solutions-1

Section 4.3-solutions-1 - troyer(lmt836 – Section 4.3 –...

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Unformatted text preview: troyer (lmt836) – Section 4.3 – isaacson – (55826) 1 This print-out should have 6 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- 18) 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 = 2 3. local maximum at x = 3 4. local maximum at x =- 2 , 3 5. local maximum at x =- 2 correct Explanation: At a local maximum of f , the derivative f ′ ( x ) will be zero, i.e. , 3( x- 3)( x + 2) parenleftBig 1 + g ( x ) 2 parenrightBig = 0 . Thus the critical points of f occur only at x =- 2 , 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 + 2) of the other two factors in f ′ ( x ). Now the sign chart- 2 3 + +- for 3( x- 3)( x + 2) shows that the graph of f is increasing on (-∞ ,- 2), decreasing on (- 2 , 3), and increasing on (3 , ∞ ). Thus f has a local maximum at x =- 2 . 002 10.0 points Use the graph a b c of the derivative of f to locate the critical points x at which f has a local maximum?...
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Section 4.3-solutions-1 - troyer(lmt836 – Section 4.3 –...

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