calc_tutorial_14_8_004

 
Problem Statement

{3,5,7}
{2,4,8}
4/2
round((pow(5,2))/(pow(4,2))+1,2)
round(2/(pow(2.56,0.5)),2)
round(5*1.25/4,2)
(5*1.56+4*1.25)

Find the minimum and maximum values of the function subject to the given constraint.

f(x, y) = 5x + 4y, x2 + y2 = 4

 
Step 1

Question Sequence

Question 1

The method of Lagrange multipliers is a general method for solving optimization problems with constraints. The steps are generally to write out the Lagrange equations, solve the Lagrange multiplier λ in terms of x and y, solve for x and y using the constraint equation and then calculate the critical values.

The constraint curve is g(x, y) = 0. Thus, g(x, y) = x2 + y2 + .

Form the Lagrange Condition ∇f = λ∇g.

<, > = λ · <2x, 2y>

Write the Lagrange equations.

= 2λx

= 2λy

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

The critical points for this optimization problem will thus come from solutions to the following system of equations.

5 = 2λx

4 = 2λy

g(x, y) = x2 + y2 - 4 = 0

Solve the first two equations for λ in terms of x and y.

7 = 2λx

λ =

A.
B.

2 = 2λy

λ =

A.
B.

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 3

By setting λ = λ, we have the equation .

We now have a system of two equations with two unknowns.

Choose to solve the first equation for x in terms of y.

A.
B.
C.
D.

Next substitute this into the constraint equation to get one equation in terms of the one unknown y.

Round your answer to two decimal places.

y2 - 4 = 0

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 4

Find the y-coordinates of the critical points. Round your answers to two decimal places.

2.56y2 - 4 = 0

- and

Find the x-coordinates of the critical points. Round your answers to two decimal places.

- and

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 5

Thus the two critical points are (−1.56, -1.25) and (1.56, 1.25). (Note that extreme points may occur also where ∇g = <2x, 2y> = (0, 0). However, the point (0, 0) is not on the constraint.)

Using the values determined above, calculate the critical values of f(x, y) = 5x + 4y. Round your answers to two decimal places.

f(-1.56, -1.25) =

f(1.56, 1.25) =

State fmin, the minimum value of f(x, y), and fmax, the maximum value of f(x, y). Round your answers to two decimal places.

fmin =

fmax =

Correct.
Incorrect.