Question 1

Lagrangian method for optimization may be used for even unconstrained two variable optimization problems.

Accepted Answer

A) True 
 B)False  False

Question 2

The local maximum for the function $f(X)=-4 X_{2}+4 X-3$ is obtained when $X$ is equal to:

Accepted Answer

A)  8 
B)  4 
C)  0.75 
D)  0.5 
A)  8 
B)  4 
C)  0.75 
D)  0.5 D

Question 3

In single-variable, constrained minimization problems, the optimal solution may be at one of the extreme PPints, thatsejs, a point where the function intersects a constraint.

Accepted Answer

A) True 
 B)False  True

Question 4

In an unconstrained two-variable problem with a quadratic objective function, there will always be a local optimal solution, though global optimal solution may not be available even in such problems.

Accepted Answer

A) True 
 B)False

Question 5

Necessary and sufficient conditions for the existence of a local maximum in a single- variable, unconstrained, nonlinear optimization problem are that the first derivative be 0 at a point and the second derivative be negative at the same point.

Accepted Answer

A) True 
 B)False

Question 6

the function $\mathrm{f}$.
$D=\frac{\partial^{2} f}{\partial X_{1}^{2}} \cdot \frac{\partial^{2} f}{\partial X_{2}^{2}}-\left(\frac{\partial^{2} f}{\partial X_{1} \partial X_{2}}\right)^{2}<0$

Accepted Answer

A) True 
 B)False

Question 7

In an unconstrained two-variable problem with a quadratic objective function, if there is a local optimum, it must also be the global optimum solution.

Accepted Answer

A) True 
 B)False

Question 8

In single-variable, unconstrained minimization problems, if there is only one local minimum, then it must be the global minimum.

Accepted Answer

A) True 
 B)False

Question 9

The necessary condition for optimality in a two-variable unconstrained function is that

Accepted Answer

A)  both partial derivatives must be non-negative 
B)  at least one of the partial derivatives must be negative 
C)  the partial derivatives must be equal and negative 
D)  both partial derivatives must be negative 
A)  both partial derivatives must be non-negative 
B)  at least one of the partial derivatives must be negative 
C)  the partial derivatives must be equal and negative 
D)  both partial derivatives must be negative

Question 10

If a local optimal solution is found for a two-variable maximization problem, the maximum value of the

Accepted Answer

A) True 
 B)False

Question 11

In single-variable, constrained minimization problems, the optimal solution will always be at a local minimum.

Accepted Answer

A) True 
 B)False

Question 12

The Lagrangian function corresponding to the following constrained optimization problem: Maximize: $f\left(X_{1}, X_{2}\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160$, subject to $X_{1}+X_{2}=10$ is

Accepted Answer

A)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}\right)$ 
B)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda .+\left(X_{1}+X_{2}\right)$ 
C)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}-10\right)$ 
D)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160-\lambda \cdot\left(X_{1}+X_{2}+10\right)$ 
A)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}\right)$ 
B)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda .+\left(X_{1}+X_{2}\right)$ 
C)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}-10\right)$ 
D)  $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160-\lambda \cdot\left(X_{1}+X_{2}+10\right)$

Question 13

Which of the following is true of the function $f(X)=16+4 X-2 X_{2}$ ?

Accepted Answer

A)  It reaches global maximum at $X=4$ 
B)  It reaches global minimum at $X=1$ 
C)  It reaches local maximum at $X=1$ 
D)  It reaches local minimum at $\mathrm{X}=1$ 
A)  It reaches global maximum at $X=4$ 
B)  It reaches global minimum at $X=1$ 
C)  It reaches local maximum at $X=1$ 
D)  It reaches local minimum at $\mathrm{X}=1$

Question 14

In an unconstrained two-variable problem with a quadratic objective function, the constant affects the value of the objective function corresponding to the optimal solution, if any, but does not affect the optimal value of the variables.

Accepted Answer

A) True 
 B)False

Question 15

Causes of nonlinearity include all of the following except

Accepted Answer

A)  additivity may be absent because mixing one cup each of two ingredients may produce less than 2 cups of products 
B)  increasing sales may result in less than proportionate total sales revenue because of price breaks 
C)  doubling of sales volume doubles sales revenue 
D)  economies produced in consumption of raw materials, such as making a 4-pack, uses less materials than individual wraps 
A)  additivity may be absent because mixing one cup each of two ingredients may produce less than 2 cups of products 
B)  increasing sales may result in less than proportionate total sales revenue because of price breaks 
C)  doubling of sales volume doubles sales revenue 
D)  economies produced in consumption of raw materials, such as making a 4-pack, uses less materials than individual wraps

Question 16

A point on a complex curve of a two-variable unconstrained function where two partial derivatives are zero is a local maximum if

Accepted Answer

A)  the second partial derivatives and mixed partial derivative are zero 
B)  the second partial derivative is negative 
C)  the second partial derivatives are negative, and the mixed partial derivative is positive 
D)  the second and mixed partial derivatives are positive 
A)  the second partial derivatives and mixed partial derivative are zero 
B)  the second partial derivative is negative 
C)  the second partial derivatives are negative, and the mixed partial derivative is positive 
D)  the second and mixed partial derivatives are positive

Question 17

We hire today a firm specializing in temporary employment opportunities in New Orleans, after the damages of Hurricane Katrina, which has developed a profit function for November 2005 given by $P=-0.25 X^{2}+100 X+5000$, where $\mathrm{X}$ is the number of employees hired in November 2005 . What should be their target employment for November in order to maximize their profit for November? Their current facilities have sufficient capacity to accommodate, at most, 400 new-hires in November 2005.

Accepted Answer

First we will solve the unconstrained pr

Question 18

In general, the values of global maximums of the objective function for unconstrained, nonlinear optimization problems will be greater than all local maximums.

Accepted Answer

A) True 
 B)False

Question 19

Nonlinear models involve more computing burden than linear models for problems of comparable size.

Accepted Answer

A) True 
 B)False

Question 20

Lagrangian method with a single $\lambda$ may be used to find optimal solutions for problems with, at most, two constraints.

Accepted Answer

A) True 
 B)False

Lagrangian method for optimization may be used for even unconstrained two variable optimization problems.

The local maximum for the function $f(X)=-4 X_{2}+4 X-3$ is obtained when $X$ is equal to:

In single-variable, constrained minimization problems, the optimal solution may be at one of the extreme PPints, thatsejs, a point where the function intersects a constraint.

In an unconstrained two-variable problem with a quadratic objective function, there will always be a local optimal solution, though global optimal solution may not be available even in such problems.

Necessary and sufficient conditions for the existence of a local maximum in a single- variable, unconstrained, nonlinear optimization problem are that the first derivative be 0 at a point and the second derivative be negative at the same point.

the function $\mathrm{f}$ . $D=\frac{\partial^{2} f}{\partial X_{1}^{2}} \cdot \frac{\partial^{2} f}{\partial X_{2}^{2}}-\left(\frac{\partial^{2} f}{\partial X_{1} \partial X_{2}}\right)^{2}<0$

In an unconstrained two-variable problem with a quadratic objective function, if there is a local optimum, it must also be the global optimum solution.

In single-variable, unconstrained minimization problems, if there is only one local minimum, then it must be the global minimum.

The necessary condition for optimality in a two-variable unconstrained function is that

If a local optimal solution is found for a two-variable maximization problem, the maximum value of the

In single-variable, constrained minimization problems, the optimal solution will always be at a local minimum.

The Lagrangian function corresponding to the following constrained optimization problem: Maximize: $f\left(X_{1}, X_{2}\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160$ , subject to $X_{1}+X_{2}=10$ is

Which of the following is true of the function $f(X)=16+4 X-2 X_{2}$ ?

In an unconstrained two-variable problem with a quadratic objective function, the constant affects the value of the objective function corresponding to the optimal solution, if any, but does not affect the optimal value of the variables.

Causes of nonlinearity include all of the following except

A point on a complex curve of a two-variable unconstrained function where two partial derivatives are zero is a local maximum if

In general, the values of global maximums of the objective function for unconstrained, nonlinear optimization problems will be greater than all local maximums.

Nonlinear models involve more computing burden than linear models for problems of comparable size.

Lagrangian method with a single $\lambda$ may be used to find optimal solutions for problems with, at most, two constraints.

Introduction to Management Science, Modeling, and Excel Spreadsheets

Forecasting

Linear Programming: Basic Concepts and Graphical Solutions

Linear Programming: Applications and Solutions

Linear Programming: Sensitivity Analysis, Duality, and Specialized Models

Transportation, Assignment, and Transshipment Problems

Integer Programming

Network Optimization Models

Multi-Criteria Models

Decision Theory

Markov Analysis

Waiting Line Models

Simulation Cdrom Modules

Filters

Exam 9: Nonlinear Optimization Models

Lagrangian method for optimization may be used for even unconstrained two variable optimization problems.

The local maximum for the function f(X)=−4X2+4X−3f(X)=-4 X_{2}+4 X-3f(X)=−4X2​+4X−3 is obtained when XXX is equal to:

In single-variable, constrained minimization problems, the optimal solution may be at one of the extreme PPints, thatsejs, a point where the function intersects a constraint.

In an unconstrained two-variable problem with a quadratic objective function, there will always be a local optimal solution, though global optimal solution may not be available even in such problems.

Necessary and sufficient conditions for the existence of a local maximum in a single- variable, unconstrained, nonlinear optimization problem are that the first derivative be 0 at a point and the second derivative be negative at the same point.

In an unconstrained two-variable problem with a quadratic objective function, if there is a local optimum, it must also be the global optimum solution.

In single-variable, unconstrained minimization problems, if there is only one local minimum, then it must be the global minimum.

The necessary condition for optimality in a two-variable unconstrained function is that

If a local optimal solution is found for a two-variable maximization problem, the maximum value of the

In single-variable, constrained minimization problems, the optimal solution will always be at a local minimum.

Which of the following is true of the function f(X)=16+4X−2X2f(X)=16+4 X-2 X_{2}f(X)=16+4X−2X2​ ?

In an unconstrained two-variable problem with a quadratic objective function, the constant affects the value of the objective function corresponding to the optimal solution, if any, but does not affect the optimal value of the variables.

Causes of nonlinearity include all of the following except

A point on a complex curve of a two-variable unconstrained function where two partial derivatives are zero is a local maximum if

In general, the values of global maximums of the objective function for unconstrained, nonlinear optimization problems will be greater than all local maximums.

Nonlinear models involve more computing burden than linear models for problems of comparable size.

Lagrangian method with a single λ\lambdaλ may be used to find optimal solutions for problems with, at most, two constraints.

Introduction to Management Science, Modeling, and Excel Spreadsheets

Forecasting

Linear Programming: Basic Concepts and Graphical Solutions

Linear Programming: Applications and Solutions

Linear Programming: Sensitivity Analysis, Duality, and Specialized Models

Transportation, Assignment, and Transshipment Problems

Integer Programming

Network Optimization Models

Multi-Criteria Models

Decision Theory

Markov Analysis

Waiting Line Models

Simulation Cdrom Modules

Filters

The local maximum for the function $f(X)=-4 X_{2}+4 X-3$ is obtained when $X$ is equal to:

Which of the following is true of the function $f(X)=16+4 X-2 X_{2}$ ?

Lagrangian method with a single $\lambda$ may be used to find optimal solutions for problems with, at most, two constraints.