Chapter 1. calc_tutorial_14_8_011-2

1.1 Problem Statement

{3}
{2}
{4}
{10,11,12,13,14,15,16,17,18,19}
12*$a/(2*$c)
round(41/27,3)
round(3*pow($d*3/41, 0.5),3)
round(pow($d*3/41, 0.5),3)
round(2*pow($d/123, 0.5),3)
round($a*$g+$b*$h+$h*$i,3)

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

f(x, y, z) = $ax + $by + $cz, x2 + 2y2 + 6z2= $d

1.2 Step 1

Question Sequence

Question 1.1

Recall the method of Lagrange Multipliers. 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 for the Lagrange multiplier λ, solve for one or more variables using the constraint equation and then calculate the critical values.

The constraint is g(x, y, z) = 0, where g(x, y, z) = x2 + 2y2 + 6z2 + iXKaZpK/a+Q=.

Form the Lagrange Condition ∇f = λ∇g.

<nc1ItEz0kR4=, iSba6t70dtA=, SFgqQUkJGdg=> = λ · <2x, 4y, 12z>

Write the Lagrange equations.

$a = XvVM00l89Is=λx

$b = h4XZagboIgc=λy

$c = DDH6Tw1RFEk=λy

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 = 4λx

$c = 12λx

x2 + 2y2 + 6z2 - $d = 0

Solve the first equation for λ.

$a = 2λx

λ =

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

We will choose to eliminate λ from $b = 4λy by substituting this expression for λ in order to express x in terms of y and also substitute this expression for λ in $c = 12λz to express x in terms of z.

x = nc1ItEz0kR4=y

x = 5dWJhLEUU4Y=z

x2 + 2y2 + 6z2 - $d = 0

Solve the first equation for y in terms of x and the second equation for z in terms of x. Then substitute these two equations into the constraint equation appropriately to get one equation in the variable x.

U2GIbglD1oM= = 0

Round the following answer to three decimal places.

or6dmYWrEbA=x2 - $d = 0

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Find the x-coordinates of the critical points. (Round your answers to three decimal places.)

$fx2 - $d = 0

x = mpgA8Zi4UKU= and -mpgA8Zi4UKU=.

Find the corresponding y-values of the critical points. (Round your answers to three decimal places.)

y = Hs0oCnDH0eo= and -Hs0oCnDH0eo=.

Find the corresponding z-values of the critical points. (Round your answers to three decimal places.)

z = zDsgB75cNfg= and -zDsgB75cNfg=.

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Thus the two critical points are (-$g, -$h, -$i) and ($g, $h, $i).

Calculate the critical values of f(x, y, z) = $ax + $by + $cz. Round your answers to three decimal places.

f(-$g, -$h, -$i) = rIjMDQwRxM4=

f($g, $h, $i) = bEz/iYnXpnk=

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

fmin = rIjMDQwRxM4=

fmax = bEz/iYnXpnk=

Correct.
Incorrect.