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Management Science Statements

Category: Business > Management
Subject: Management Science
Details: 1. Which of the following statements are true for f(x) = x2? (Chapter 10)

a. f(x) is a concave function
b. f(x) </= 20 is a convex set
c. f(x) >/= 5 is a convex set
d. none of the above

2. Which of the following statements are true for f(x) = 3x2 + 2x + 5? (Chapter 10)

a. f(x) is not a concave function
b. f(x) </= 10 is a convex set
c. f(x) > /= 2 is a convex set
d. all of the above

3. Which of the following statements are true for an unconstrained optimization problem? (Chapter 10)

a. At a global min or max all partial derivatives will be equal to 0.
b. The first-order optimally conditions are used to find the local maxima and
local minima.
c. At a local min or max all partial derivatives will be equal to 1.
d. none of the above

4. The Lagrange multiplier _______________. (Chapter 10)

a. is the shadow price
b. in not valid over a range of changes in the RHS
c. is the rate of change in the objective value as the RHS of the constraint
increases
d. all the above

5. The optimal value function of a portfolio analysis problem solved using quadratic
programming is __________________. (Chapter 10)

a. called the efficient frontier
b. hurt when an active constraint is tightened
c. a piecewise quadratic convex function
d. all the above

6. Increasing the RHS on a nonlinear </= constraint __________. (Chapter 10)

a. can contract the constraint set
b. may contract the constraint set
c. may hurt the optimal objective value
d. none of the above

Solution Preview

1. Which of the following statements are true for f(x) = x2? (Chapter 10)
<br>
<br>a. f(x) is a concave function
<br>b. f(x) &lt;/= 20 is a convex set
<br>c. f(x) &gt;/= 5 is a convex set
<br>d. none of the above
<br>Answer b: A set S is called a convex set if the line segment joining any pair of points of S lies entirely in S. Consider 2 points -(20)0.5, (20)0.5. If we square -(20)0.5, (20)0.5 then we get (20,20), both of these are &lt;/= 20 (this matches the requisites for part b). Now, we check if they are convex. Let us arbitrarily take a linear combination of -(20)0.5, (20)0.5 let us say (-(20)0.5+(20)0.5)/2 or 0. Now, let us square 0 and we get 0. Is 0 &lt;/= 20? Yes it is, hence part b) is right.
<br>2. Which of the following statements are true for f(x) = 3x2 + 2x + 5? (Chapter 10)
<br>
<br>a. f(x) is not a concave function
<br>b. f(x) &lt;/= 10 is a convex set
<br>c. f(x) &gt; /= 2 is a convex set
<br>d. all of ...

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