1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
eimsori [14]
3 years ago
11

This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the ex

treme values of the function subject to the given constraint. f(x, y, z) = 8x + 8y + 4z; 4x2 + 4y2 + 4z2 = 36 maximum value minimum value
Mathematics
2 answers:
velikii [3]3 years ago
7 0

Answer:

The maximum value= 36

Minimum value = - 36

Step-by-step explanation:

Given that

f(x, y, z) = 8 x + 8 y + 4 z

h(x,y,z)=4 x² + 4 y² + 4 z² - 36

From Lagrange multipliers

Δf = λ Δh

Δf = < 8 ,8 , 4>

Δh = < 8 x ,8 y  , 8 z>

Δf = λ Δh

So

< 8 ,8 , 4> = < 8  λ x ,8 λ y  , 8 λ z>

8 = 8  λ x                     -------------1

8 = 8 λ y                      ----  ------2

4 = 8 λ z                    ----------------3

From equation 1 ,2 and 3

Now by putting the value of x,y and z in the following equation

4 x² + 4 y² + 4 z² = 36

4\times \dfrac{1}{\lambda^2 }+4\times \dfrac{1}{\lambda^2 }+4\times \dfrac{1}{(2\lambda)^2 }=36

\dfrac{4}{\lambda^2 }+ \dfrac{4}{\lambda^2 }+ \dfrac{1}{\lambda^2 }=36

So the value of λ is

\lambda =\pm \dfrac{1}{2}

When λ = 1/2

x = 1 / λ   , y=1 / λ   ,  z= 1 /2 λ

x= 2 , y = 2 , z=1

So

f(x, y, z) = 8 x + 8 y + 4 z

f(2, 2, 1) = 8 x 2 + 8 x 2 + 4 x 1

f(2, 2, 1) =36

When λ = - 1/2

x = 1 / λ   , y=1 / λ   ,  z= 1 /2 λ

x= - 2 , y = - 2 , z= - 1

So

f(x, y, z) = 8 x + 8 y + 4 z

f(-2, -2, -1) = 8 x (-2) + 8 x (-2) + 4 x (-1)

f(-2, -2, -1) = - 36

The maximum value= 36

Minimum value = - 36

Taya2010 [7]3 years ago
3 0

Answer:

Maximum value of f(x,y,z)=36 at (2,2,1)

Minimum value of f(x,y,z)=-36 at (-2,-2,-1)

Step-by-step explanation:

We are given that

f(x,y,z)=8x+8y+4z

g(x,y,z)=4x^2+4y^2+4z^2=36

We have to find the extreme values of the function using Lagrange multipliers.

f_x(x,y,z)=8

f_y(x,y,z)=8

f_z(x,y,z)=4

g_x(x,y,z)=8x

g_y(x,y,z)=8y

g_z(x,y,z)=8z

f_x=\lambda g_x

8=8x\lambda

x=\frac{1}{\lambda}

f_y=\lambda g_y

8=8y\lambda

y=\frac{1}{\lambda}

f_z=\lambda g_y

4=8z\lambda

z=\frac{1}{2\lambda}

Substitute the values in g(x,y,z)

4(\frac{1}{\lambda})^2+4(\frac{1}{\lambda})^2+4(\frac{1}{2\lambda})^2=36

\frac{4}{\lambda^2}+\frac{4}{\lambda^2}+\frac{1}{\lambda^2}=36

\frac{9}{\lambda^2}=36

\lambda^2=\frac{9}{36}=\frac{1}{4}

\lambda=\pm\frac{1}{2}

Substitute \lambda=\frac{1}{2}

x=2,y=2,z=1

Substitute \lambda=-\frac{1}{2}

x=-2,y=-2,z=-1

Now, f(2,2,1)=8(2)+8(2)+4(1)=16+16+4=36

f(-2,-2,-1)=8(-2)+8(-2)+4(-1)=-16-16-4=-36

Maximum value of f(x,y,z)=36 at (2,2,1)

Minimum value of f(x,y,z)=-36 at (-2,-2,-1)

You might be interested in
What is (a+2)(a+2) make sure to multiply
a_sh-v [17]

Answer:

a² + 4 a + 4

Step-by-step explanation:

( a + 2 ) ( a + 2 )

Expand brackets

a² + 2 a + 2 a + 4

Simplify

a² + 4 a + 4

7 0
2 years ago
Read 2 more answers
Find the missing angle measure. then classify the triangle by its angles and by its sides.
iren [92.7K]

Answer:

Part a) The measure of the missing angle is 90\°

Part b) The triangle of the figure is a right triangle

Part c) The triangle of the figure is a scalene triangle

Step-by-step explanation:

we know that

The sum of the interior angles of a triangle is equal to 180\°

so

Let

x------> the missing angle

we know that

x\°+50\°+40\°=180\°

solve for x

x=180\°-90\°=90\°

The triangle of the figure is a right triangle --------> by its angles

Because the triangle has an angle measure of 90\°

The triangle of the figure is a scalene triangle --------> by its sides

Because the three angles and the three sides measures are different

8 0
3 years ago
Read 2 more answers
Sara is making a key chain using the bead design shown. What fraction of the beads in her design are either blue or red? 
klio [65]
You can make a ratio of red to overall colors and then that will give you the fraction, you might have to simplify it
4 0
3 years ago
If a phone cost $224 what would it cost if there was a 7% sales tax.
Annette [7]

Answer:

<u>$239.68</u>

Step-by-step explanation:

<u>Given</u>

  • Cost = $224
  • Sales Tax = 7%

<u>Solving</u>

  • Cost w/ Tax = Cost (1 + Tax)
  • Cost w/ Tax = 224 (1 + 0.07)
  • Cost w/ Tax = 224 * 1.07
  • Cost w/ Tax = <u>$239.68</u>
5 0
2 years ago
What is g(x)? (apex algebra 1 semester 2 2019).
Contact [7]

Answer:

D. g(x)=-2^x

Step-by-step explanation:

We can use a process of elimination in order to easily solve this.

The shape of g(x)=-|x| will be two diagonal lines that meet at (0,0)

g(x)=-x^2 Will be an upside down parabola

g(x)=-x Will be a line with a slope of -1.

This means that the answer must be D

6 0
3 years ago
Read 2 more answers
Other questions:
  • Plz help with this!!
    7·1 answer
  • A circle is described by the equation x2 + y2 + 14x + 2y + 14 = 0. What are the coordinates for the center of the circle and the
    13·1 answer
  • Need the answer quick!
    13·1 answer
  • I need help i'm never good at math :((
    7·1 answer
  • Se A {a, b, c, d, e, f} e B = {a, e, i, o, u}. Quantos elementos possui ∩ ?l
    13·1 answer
  • 23= 7/9(6x-36)+9<br><br> solve for x<br><br> x=__
    6·2 answers
  • Mr. Thomas cooked 36 hamburgers and hot dogs at a cookout. He cooked twice as many hot dogs as hamburgers. How many hamburgers a
    5·2 answers
  • Identify the point on the number line for each number
    5·1 answer
  • Write the recurring decimal 0.1 as a fraction in its simplest form
    12·2 answers
  • Find the inverse of each function. Is the inverse a function? f(x)=x⁷
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!