HW5 Solutions - ASSIGNMENT 5 - SOLUTIONS Problem 1. Done in...

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ASSIGNMENT 5 - SOLUTIONS Problem 1. Done in class. Problem 2. Take f ( x ) = x 2 , whose graph is a parabola. Drop it by 1 (so now the function is g ( x ) = x 2 - 1). Slide it to the right by 1 (the function becomes h ( x ) = ( x - 1) 2 - 1 = x 2 - 2 x .) This graph has all the required properties. Problem 3. By definition, concave upwards means increasing first derivative. If f Í and g Í are increasing, then so is their sum f Í + g Í = ( f + g ) Í . So the first statement is true. But the second is false. For example, both f ( x ) = x 2 and g ( x ) = x 4 + x 2 are concave upwards on [0 , 1] yet f ( x ) - g ( x ) = - x 4 is concave downwards on [0 , 1]. Problem 4. To start off, the domain of f ( x ) = x 9 - x is ( -∞ , 9]. The first derivative is f Í ( x ) = x Í 9 - x + x ( 9 - x ) Í = 9 - x + x 1 2 1 9 - x (9 - x ) Í = 9 - x - x 2 9 - x = 2(9 - x ) - x 2 9 - x = 18 - 3 x 2 9 - x = 3 2 6 - x 9 - x The critical numbers of f ( x ) are x = 6 ( f Í (6) = 0) and
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This note was uploaded on 07/31/2011 for the course MATH 102 taught by Professor Maryamnamazi during the Spring '10 term at University of Victoria.

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