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Unformatted text preview: (.5)) = 15<0. Thus f(x 1 , x 2 , x 3 ) is concave on R 3 . 12. We know that (1) f(cx + (1  c)y) cf(x) + (1  c)f(y) (2) g(cx + (1  c)y) cg(x) + (1  c)g(y) Adding (1) and (2) yields h(cx + (1  c)y) ch(x) + (1  c)h(y) which shows that h is also a convex function. 13. Since f is convex we know that for 0 k 1 (1) f(kx + (1  k)y) kf(x) + (1  k)f(y). Multiplying both sides of (1) by c 0 shows that g is also a convex function....
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 Spring '07
 HOCHBAUM
 Operations Research

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