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
alekssr [168]
3 years ago
7

Find the minimum and maximum of f(x,y,z)=x^2+y^2+z^2 subject to two constraints, x+2y+z=4 and x-y=8.

Mathematics
1 answer:
Alika [10]3 years ago
7 0
The Lagrangian for this function and the given constraints is

L(x,y,z,\lambda_1,\lambda_2)=x^2+y^2+z^2+\lambda_1(x+2y+z-4)+\lambda_2(x-y-8)

which has partial derivatives (set equal to 0) satisfying

\begin{cases}L_x=2x+\lambda_1+\lambda_2=0\\L_y=2y+2\lambda_1-\lambda_2=0\\L_z=2z+\lambda_1=0\\L_{\lambda_1}=x+2y+z-4=0\\L_{\lambda_2}=x-y-8=0\end{cases}

This is a fairly standard linear system. Solving yields Lagrange multipliers of \lambda_1=-\dfrac{32}{11} and \lambda_2=-\dfrac{104}{11}, and at the same time we find only one critical point at (x,y,z)=\left(\dfrac{68}{11},-\dfrac{20}{11},\dfrac{16}{11}\right).

Check the Hessian for f(x,y,z), given by

\mathbf H(x,y,z)=\begin{bmatrix}f_{xx}&f_{xy}&f_{xz}\\f_{yx}&f_{yy}&f_{yz}\\f_{zx}&f_{zy}&f_{zz}\end{bmatrix}=\begin{bmatrix}=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}

\mathbf H is positive definite, since \mathbf v^\top\mathbf{Hv}>0 for any vector \mathbf v=\begin{bmatrix}x&y&z\end{bmatrix}^\top, which means f(x,y,z)=x^2+y^2+z^2 attains a minimum value of \dfrac{480}{11} at \left(\dfrac{68}{11},-\dfrac{20}{11},\dfrac{16}{11}\right). There is no maximum over the given constraints.
You might be interested in
Please explain step by step worth 20 points
ankoles [38]
Multipy $275 by the 4.5%
4.5%=.045 (just move the decimal 2 places left)
275*.045= 12.375  Susan gets $12.375 a year (don't round yet)
Multiply $12.375 by 5years          12.375*5= 61.875 (now round)
$61.90=B
I hope this helps! :)

5 0
3 years ago
A ladder is leaning up against the side of a house. Use two points
Natali [406]

Answer:

The slope is 2.

Step-by-step explanation:

Slope = rise / run, meaning that when you move 2 units up, you move 1 unit right. The slope here is positive.

4 0
2 years ago
Yoko is buying a car from a local car dealership. She wants to get the lowest interest rate possible. Which will most likely hel
Svetach [21]

Answer: The answer is (b) making sure she has a positive credit history.

Step-by-step explanation: Given that Yolo is buying a car from a local car dealership and wants to get the lowest interest rate possible. We are to select the correct option which will most likely help her.

From the given options, we can see that she will get the lowest interest rate possible if she has a positive credit score.

Thus, (b) is the correct option.

9 0
3 years ago
Read 2 more answers
HELP PLEASE <br><br>must show work <br><br>I have the answer just need to show work​
Kazeer [188]

Answer:

Step-by-step explanation:

To solve these equations involving variables and exponents we need to follow these steps.

1) We need to find out the factor that is common in the equation.

2) After taking common, solve the equation. We can add or subtract only those values that have same bases.

1) 8+6x^4

here we can see, both numbers are divisible by 2, so taking 2 common

=2(8/2 + 6x^4/2)\\= 2(4 + 3x^4)

It cannot be further simplified because both number donot have same bases.

3.4n^9 + 12 n

We can take 4n common

=4n(4n^9/4n + 12 n/4n)\\=4n(n^8 + 3)

5. -12a -3

Here -3 cam be taken common

= -3(-12a/-3 -3/-3)

= -3(4a +1)

7. 12n^5 + 16n^3

here the smallest power of n is n^3 so, we can take n^3 common and both coefficients are divisible by 4 so taking 4n^3 common

4n^3( 3n^2 + 4)

9. 5k^2 - 40k+10

Here we cannot take k common, as k is not a multiple of 10. For taking common it should be divisible by each value in the equation. But each value s divisible by 5 so, taking 5 common

=5(k^2 - 8k + 2)

11.-60 + 60n^2 +50n^3

Here we cannot take n common, as n is not a multiple of -60. For taking common it should be divisible by each value in the equation. But each value s divisible by 10 so, taking 10 common

=10(-6 + 6n^2 +5n^3)

13. -36n^3 -12n-28

Here we cannot take n common, as n is not a multiple of 28. For taking common it should be divisible by each value in the equation. But each value s divisible by -4 so, taking -4 common

=-4(9n^3 + 3n +7)

15. 63n^3+81n+18

Here we cannot take n common, as n is not a multiple of 18. For taking common it should be divisible by each value in the equation. But each value s divisible by 9 so, taking 9 common

=9(7n^3 + 9n + 2)

17. -24a^2b^2 + 36ab-60a

=6a(-4ab^2+6b-10)

3 0
3 years ago
A scientist writes the equation n(h)100e^0.25h to model the growth of a certain bacteria in a petri dish, where N represents the
Natasha2012 [34]

Answer: Second option is correct.

Step-by-step explanation:

Since we have given that

n(h)=100e^{0.25h}

where, N represents the number of bacteria after 'h' hours.

So, we have given that there are 450 bacteria present.

N = 450

According to question,

450=100e^{0.25h}\\\\\frac{450}{100}=e^{0.25h}\\\\4.5=e^{0.25h}\\\\\text{Taking natural log 'ln' on both sides}\\\\\ln (4.5)=0.25h\\\\1.50=0.25h\\\\h=\frac{1.50}{0.25}\\\\h\approx 6\ hours

Hence, Second option is correct.

8 0
3 years ago
Read 2 more answers
Other questions:
  • Which composition of transformations would be the inverse of rotating a figure 90° counterclockwise and then reflecting it over
    12·2 answers
  • A^4 + 3a² - ( 5a - 3)(-7a)<br><br> Please show work!!
    13·1 answer
  • What is 944÷14 (include the remainder)
    9·2 answers
  • How do I solve this ?
    11·2 answers
  • Which inequality is represented by this graph?
    15·2 answers
  • HELP PLEASE IM GOING TO FAIL WILL GIVE BRAINLESS
    7·1 answer
  • Help me!! 16 points for this
    15·2 answers
  • On a loan of $4,500 for 2 1/2 years at 4% per year, how much do you wind up paying to pay off the loan?
    12·2 answers
  • 2) State the area of the given triangle.
    13·2 answers
  • Round 2593.6781 to nearest thousandths.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!