Problems 41-48 refer to the bounded feasible region with corner points O = 0 , 0 , A = 0 , 5 , B = 4 , 3 , and C = 5 , 0 that is determined by the system of inequalities x + 2 y ≤ 10 3 x + y ≤ 15 x , y ≥ 0 If P = a x + 10 y , find all numbers a such that the maximum value of P occurs at both B and C .
Problems 41-48 refer to the bounded feasible region with corner points O = 0 , 0 , A = 0 , 5 , B = 4 , 3 , and C = 5 , 0 that is determined by the system of inequalities x + 2 y ≤ 10 3 x + y ≤ 15 x , y ≥ 0 If P = a x + 10 y , find all numbers a such that the maximum value of P occurs at both B and C .
Solution Summary: The author calculates that the maximum value of the objective function P=ax+10y occurs at both B and C if the bounded feasible region has the corner points.
Problems 41-48 refer to the bounded feasible region with corner points
O
=
0
,
0
,
A
=
0
,
5
,
B
=
4
,
3
, and
C
=
5
,
0
that is determined by the system of inequalities
x
+
2
y
≤
10
3
x
+
y
≤
15
x
,
y
≥
0
If
P
=
a
x
+
10
y
, find all numbers
a
such that the maximum value of
P
occurs at both
B
and
C
.
Prove let Aand B submodul of M
A is large sub podule A large of B
and B large of M.
SM
B Smale sub module B/A smal of M/A
and As Mallof M.
Give example and expleain caim.
Amonorphism and split
d) Determine the following group: Hom, (Q,Z)
and Ho M₂ (Q, Q) and Hom (2/12, Q) =
Q2: Using the Laplace transform, find the solution for the following equation
y"" +y" = 6et + 6t + 6. Suppose zero initial conditions (y"" (0) = y"(0) = y'(0) = y(0) = 0).
1- Let A = {A1, A2, ...), in which A, A, = 0, when i j.
a) Is A a π-system? If not, which element(s) should be added to A to become a π-system?
b) Prove that σ(A) consists of the finite or countable unions of elements of A; i.c., A E σ(A) if and
only if there exists finite or countable sequence {n} such that A = U₁An (Hint: Let F be such
class; prove that F is a σ-filed containing A.)
c) Let p ≥ 0 be a sequence of non-negative real numbers with Σip₁ = 1. Using p₁'s, how do you
construct a probability measure on σ(A)? (Hint: use extension theorem.)
2- Construct an example for which P(lim sup A,) = 1 and P(lim inf An) = 0.
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