Chapter 1. calc_tutorial_14_8_004

1.1 Problem Statement

{3,5,7}
{2,4,8}
$b/2
round((pow($a,2))/(pow($b,2))+1,2)
round(2/(pow($d,0.5)),2)
round($a*$e/$b,2)
($a*$f+$b*$e)

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

f(x, y) = $ax + $by, x2 + y2 = 4

1.2 Step 1

Question Sequence

Question 1.1

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 + sWNiRN4ZHUU=.

Form the Lagrange Condition ∇f = λ∇g.

<nc1ItEz0kR4=, iSba6t70dtA=> = λ · <2x, 2y>

Write the Lagrange equations.

nc1ItEz0kR4= = 2λx

iSba6t70dtA= = 2λy

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

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

$a = 2λx

$b = 2λy

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

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

7 = 2λx

λ =

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

2 = 2λy

λ =

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

Correct.
Incorrect.

1.4 Step 2

Question Sequence

Question 1.3

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.

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

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.

U2GIbglD1oM=y2 - 4 = 0

Correct.
Incorrect.

1.5 Step 2

Question Sequence

Question 1.4

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

$dy2 - 4 = 0

-5dWJhLEUU4Y= and 5dWJhLEUU4Y=

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

-or6dmYWrEbA= and or6dmYWrEbA=

Correct.
Incorrect.

1.6 Step 2

Question Sequence

Question 1.5

Thus the two critical points are (−$f, -$e) and ($f, $e). (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) = $ax + $by. Round your answers to two decimal places.

f(-$f, -$e) = l49xo4exqbc=

f($f, $e) = mpgA8Zi4UKU=

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 = l49xo4exqbc=

fmax = mpgA8Zi4UKU=

Correct.
Incorrect.