October 1, 2007 MTHSC 974.001 - Homework # 5 1. Let f : R R satisfy lim f (x) = . Assume f is bounded below.
a. Prove the minimizing sequence must be bounded. Proof: Let f : R R. Suppose f is bounded below, non-empty and lim f (x) = .
September 19, 2007
MTHSC 974.001 - Homework # 4
1. Let f and g be convex functions in the modern sense. Let x be an interior point of the domain of f + g . Prove: (a) D (f + g )(x) D f (x) + D g (x)
Proof: Let f, g conV (R). Then let x0 int(d
September 12, 2007
MTHSC 974.001 - Homework # 3
1. Prove that if f conV(R), then its domain must be an interval. Hint: Suppose t dom(f ) and it is isolated. Then an r > 0 (t r, t) (t, t + r) dom(f ) / and s > t + r, s dom(f ). You should be a
August 31, 2007 MTHSC 974.001 - Homework # 2 1. For r > o, draw the graph of
1 f r (X ) = 1 to see if f r is convex on R.
1 2 2 rx ,
1 2r ,
1 r 1 r
1 Although the drawing is rudimentary, it appears that f r (x) is convex o
October 10, 2007
MTHSC 974.001 - Homework # 6
1. If we dene f by f (x, y ) =
(9x2 y 2 ) (3xy ) ,
if x + y = 3 if x + y = 3
(6, 1) Is f continuous at (2, 1)?
2. If f : Df Rm Rn is continuous on Df and g : Dg Rn Rr is continuous on Dg Formu