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# HW2Sol - Fall 2010 IEOR 160 Industrial Engineering...

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Fall 2010 IEOR 160 Industrial Engineering & Operations Research September 13, 2010 Page 1 of 2 HOMEWORK 2 SOLUTIONS Chapter 12.3 2. Since f 00 ( x ) = 6 x , then f 00 ( x ) > 0 for x > 0, f 00 ( x ) < 0 for x < 0, f 00 ( x ) = 0 for x = 0 So f ( x ) is neither convex nor concave. 5. f 00 ( x ) = - x - 2 < 0 for x > 0, so f ( x ) is a convex function. 10. H = 2 a b b 2 c By Th 3, the function will be convex if all the principal minors are nonnegative, i.e. 2 c 0 , a 0 and 4 ac - b 0. By Th 3 0 , the function will be concave if a 0 , 2 c 0 and 4 ac - b 2 0, which ensures that the first principal minors are nonpositive and the second principal minor is nonnegative. 12. Since f and g are convex, letting x = ( x 1 , x 2 , . . . , x n ) , y = ( y 1 , y 2 , . . . , y n ), we have that for 0 k 1 f ( kx + (1 - k ) y ) kf ( x ) + (1 - k ) f ( y ) (1) g ( kx + (1 - k ) y ) kg ( x ) + (1 - k ) g ( y ) (2) Adding (1) and (2) yields h ( kx + (1 - k ) y ) kh ( x ) + (1 - k ) h ( y ) which shows that h is also a convex function. 13. Since f is convex, letting x = ( x 1 , x 2 , . . . , x n ) , y = ( y 1 , y 2 , . . . , y n ), we have that for 0 k 1 f ( kx + (1 - k ) y ) kf ( x ) + (1 - k ) f ( y ) (3) Multiplying both sides of (3) by c 0 yields

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HW2Sol - Fall 2010 IEOR 160 Industrial Engineering...

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