Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 11.7, Problem 28PS
To determine
The absolute extrema of
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13.
y= f(x).
y= g(x)-
Graphs of the functions f and g are shown in the
xy-plane above. For which of the following values of
x does f(x) +g(x) = 0 ?
A) -3
B) -2
C) -1
D) 0
a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x)
= infyec f(x,y) is a convex function.
b. Consider the following optimization problem
min fo(x)
s.t.
Ax = b
where A E R"X" and b e R". Let x* E Xopt. Write out an equivalent condition
that characterizes x* and prove your conclusion.
c. Solve the following optimization problem by first writing out its equivalent
optimization problem:
min
c'x
s.t.
xT Ax m||x – y||3 holds for all x, y.
j. Let f (x) = ||x||. Find out its conjugate function f* and prove your conclusion.
2. Show that the following functions are convex by verifying the condition that
² f(x) 20
is satisfied for all a in the domain of f:
(a) f(u₁, ₂) In(e" + e"),
(b) f(ui, u2, U3, U₁)=-In(1-u₁-u2-uz-ua) over the domain {ue R4 | u₁+U₂+uz+us < 1}.
Chapter 11 Solutions
Calculus
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