a151q6wi

a151q6wi - F increasing Interval-∞-3-3-1-1 1(1 ∞ Test...

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MATH 151 Answers to Assignment 6 Autumn 2009 1. Find and classify the critical points of f ( x )=3 x 4 +4 x 3 - 12 x 2 into local maxima, local minima and neither. f ± ( x ) = 12 x 3 +12 x 2 - 24 x = 12 x ( x 2 + x - 2) = 12 x ( x +2)( x - 1). f is always di±erentiable. f ± ( x ) = 0 at x = 0, - 2 and 1. f ± ( x ) = 36 x 2 + 24 x - 24. f ± (0) = - 24 < 0; f ± ( - 2) = 36( - 2) 2 + 24( - 2) - 24 = 72 > 0; f ± (1) = 36(1) 2 + 24(1) - 24 = 36 > 0; Answer : x = 0 is a local maximum; x = - 2 and x = 1 are local minima. 2. Find the interval(s) in which the graph of y = x 4 +4 x 3 - 18 x 2 is concave up. y ± =4 x 3 + 12 x 2 - 36 x ; y ± = 12 x 2 + 24 x - 36 = 12( x 2 +2 x - 3) = 12( x + 3)( x - 1). Interval ( -∞ , - 3) ( - 3 , 1) (1 , ) Test Point - 4 0 2 y ± 12( - 4 + 3)( - 4 - 1) = 60 - 36 12(2 + 3)(2 - 1) = 60 Answer : ( -∞ , - 3) (1 , ) (over) Typeset using A M S -T E X.

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MATH 151 Answers to Assignment 6, p .2 Autumn 2009 3. F is a function that is deFned and continuous at all real numbers. The derivative of F is given by F ± ( x )= ( x - 1) 2 ( x + 3) 5 ± ( x + 1) 3 (i) What are the critical points of F ? F ± ( x ) = 0 at x = 1 and x = - 3. F ± does not exist at x = - 1. Answer : - 1, 1 and - 3 (ii) Where is the function
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Unformatted text preview: F increasing? Interval (-∞ ,-3) (-3 ,-1) (-1 , 1) (1 , ∞ ) Test Point-33-2 31 F ± (-33-1) 2 (-33+3) 5 √ (-33+1) 3 (-2-1) 2 (-2+3) 5 √ (-2+1) 3 (0-1) 2 (0+3) 5 √ (0+1) 3 (31-1) 2 (31+3) 5 √ (31+1) 3 = 4335 =-9 = 3 = 3845 F ± ( x ) > 0 in (-∞ ,-3), (-1 , 1) and (1 , ∞ ). Since F is continuous at x = 1, we can combine the second and third intevals. Answer : (-∞ ,-3) ∪ (-1 , ∞ ) (iii) ±or each critical point of F , decide if it is a local maximum, local minimum or neither. F ± goes from positive to negative as we cross x =-3; F ± goes from negative to positive as we cross x =-1; F ± does not change sign as we cross x = 1. Answer :-1 is a rel. min, 3 is a rel. max, and 1 is neither. end...
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a151q6wi - F increasing Interval-∞-3-3-1-1 1(1 ∞ Test...

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