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
This is the end of the preview. Sign up
access the rest of the document.
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.