The max value and min value can then be determined from these critical points.
To find the extreme values of a function subject to a constraint, we can use Lagrange multipliers. First, we set up the Lagrangian equation by multiplying the constraint by a scalar λ and adding it to the original function.
Then, we take the partial derivatives of the Lagrangian equation with respect to each variable and set them equal to zero. This will give us a system of equations to solve for the critical points.
Once we have the critical points, we need to determine which ones are maximums and which are minimums.
To do this, we can use the second derivative test. If the second derivative is positive at a critical point, it is a minimum. If the second derivative is negative, it is a maximum.
In summary, to find the extreme values of a function subject to a constraint using Lagrange multipliers, we set up the Lagrangian equation, solve for the critical points, and then use the second derivative test to determine which ones are maximums and which are minimums.
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The maximum value of f(x, y, z) is 26.5, and the minimum value is -29.
How did we get the values?To find the extreme values of the function f(x, y, z) = 6x + 6y + 5z subject to the constraint 3x² + 3y² + 5z² = 29 using Lagrange multipliers, set up the following system of equations:
1. ∇ f = λ∇g
2. g(x, y, z) = 3x² + 3y² + 5z² - 29
where ∇f and ∇g are the gradients of f and g respectively, and λ is the Lagrange multiplier.
Taking the partial derivatives, we have:
∇ f = (6, 6, 5)
∇g = (6x, 6y, 10z)
Setting these two gradients equal to each other, we get:
6 = 6λx
6 = 6λy
5 = 10λz
Dividing the first two equations by 6\(\lambda\), we obtain:
x = ¹/λ
y = ¹/λ
Substituting these values into the third equation, we have:
5 = 10λz
z = ¹/2λ
Now, substitute x, y, and z back into the constraint equation to find the value of λ:
3(¹/λ)² + 3(¹/λ)² + 5(1/2λ)² = 29
6(¹/λ²) + 5(⁴/λ²) = 29
24 + 5 = 116λ²
116λ² = 29
λ² = ²⁹/₁₁₆
λ = ±√²⁹/₁₁₆
λ = ± √²⁹/2√29
λ = ± ¹/₂
We have two possible values for λ, λ = ¹/₂ and λ = ¹/₂
Case 1: λ = ¹/₂
Using this value of λ, we can find the corresponding values of x, y, and z:
x = ¹/λ = 2
y =¹/λ = 2
z = 1/2 λ = ¹/₂
Case 2: λ = -1/2
Using this value of λ, find the corresponding values of x, y, and z:
x = 1/λ = -2
y = 1/λ = -2
z = 1/(2λ) = -1
Now that we have the values of x, y, and z for both cases, substitute them into the objective function f(x, y, z) to find the extreme values.
For Case 1:
f(x, y, z) = 6x + 6y + 5z
= 6(2) + 6(2) + 5(1/2)
= 12 + 12 + 2.5
= 26.5
For Case 2:
f(x, y, z) = 6x + 6y + 5z
= 6(-2) + 6(-2) + 5(-1)
= -12 - 12 - 5
= -29
Therefore, the maximum value of f(x, y, z) is 26.5, and the minimum value is -29.
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Write a ratio for the following situation. For every 10 toes, there are 2 feet.
Answer:
5:1
Step-by-step explanation:
10 toes = 2 feet
10:2
Simplify!
5:1
Use the ordered pairs shown in the table.
Which equation represents the data in the
table?
Answer:
Option C y = 1.5x
Step-by-step explanation:
If you look closely you will notice that there is a pattern that follows all the time. And the pattern is...
y = 1.5x
Estelle's graduation picnic will cost $15 if it has 5 attendees. If there are 26 attendees, how much will Estelle's graduation picnic cost? Solve using unit rates.
Answer:
You can try to do a direct rule of three in this exercise.
We can considerr that the relation is proportional and we know that in this case it's dorect, as the more attendees there come, the more Estelle will have to pay, so we can write this and get x:
5 attendees - 15 dollars
26 attendees - x dollars
x = 26*15 / 5 = 78 dollars
So you know how much Estelle would have to pay
Hamburgers cost $4.50 at a football stadium, write an equation that shows the total cost,c of n hamburgers?
Answer:
c=4.50
Step-by-step explanation:
c=4.50 the answer is at the top.
Answer:
N=$4.50
Step-by-step explanation:
If the price for one hamburger is $4.50, then N=$4.50
15% of 130 using the decimal method
Answer:
0.3846 is a decimal and 38.46/100 or 38.46%is the percentage for 50/130
Step-by-step explanation:
Pls mark me in brain list
Answer:
0.3846 is a decimal and 38.46/100 or 38.46%is the percentage for 50/130
Step-by-step explanation:
8p - p+0.75 = 4 + 0.25
brainliest to whoever can answer questions correctly
Answer:
v ~ 32.72
Hope that helps!
En un almacén de dimensiones 5m de largo ,3m de ancho y 2m de alto queremos almacenar cajas de dimensiones 10 dm de largo 6dm de ancho y 4 dm de alto ¿cuántas cajas podremos almacenar?
Answer:
how do you translate
Step-by-step explanation:
Helllllllp
Simplify the expression 43 + 2(3 − 2).
14
16
66
68
Answer:
68
Step-by-step explanation:
Answer:
The answer is 46...i cant find that answer in the choices though
Step-by-step explanation:
In PEMDAS we can see that we have parentheses first so we solve whatever is in the parentheses. which 3-2 = 1....so it should look like this
43 + 2 + 1
Which equals 46
hope this helps. have a good day
Given the function f(x) graphed on the coordinate plane 15
below, determine f(3).
F(3)= ?
Answer:
2
Step-by-step explanation:
A movie theater is located at (1,7) after walking 6 blocks to the movie theater you meet you friend at your friend's house. What ordered pair corresponds to the location of your friend's house
Answer:
(-5,1)
Step-by-step explanation:
Given the coordinate of the location of the theatre as (1,7)
If this was the coordinate after moving 6 blocks to your friend's house, then the coordinate of the location of my friend house will be gotten as thus;
C = (1-6, 7-6) (since he moved 6blocks)
C = (-5, 1)
Hence the required coordinate is (-5,1)
POR QUE ES IMPORTANTE LA GEOMETRIA EN NUESTRA VIDA
Step-by-step explanation:
La geometría permite a los estudiantes conectar objetos cartográficos en el aula con contextos del mundo real con respecto a la dirección y el lugar. La comprensión de las relaciones espaciales también se considera importante en el papel de la resolución de problemas y las habilidades de pensamiento de orden superior.
27 of the 180 vehicles in a movie theater parking lot are minivans.What percent of the vehicles in the parking lot are minivans?
Answer:
160
Step-by-step explanation:
Answer:
15 percent
Step-by-step explanation:
First use the equation part over total to get 27/180. Then reduce the equation by dividing both the numerator 27 and the denominator 180 by 9. You are left with 3/20. Then use your calculator or you can work it out long hand to get 0.15. You times 0.15 by 100 if you want to find the percent, leaving you with 15 percent.
I hope this helped! :)
given teach trig ratio as a fraction in simplest form:sin(N)
Answer:
Choice #1: sq rt 3/2
Step-by-step explanation:
cos N=8/PN
PN=16
tan N=PO/8
PO=13.86
sin N=13.86/16=0.866=sq rt 3/2
What is 30% of 42
I’ve given up on my homework please help
30% of 42 is 12.6.
sorry if wrong.
hope it helps.
have a great day.
Answer:
30% of 42 is 12.6.
Step-by-step explanation:
Ok, so first you need to find the value of 1%. To do that, you divide 42 by 100. That gives you 0.42. So, 0.42 = 1%. TO find 30%, multiply the value of 1% by 30. That's 0.42 * 30, which gives you 12.6. Therefore, 30% of 42 is 12.6!
Someone plz help me with this question with an explanation
To use the angle-angle similarity theorem you need to know 2 corresponding angles of each triangle are the same.
You are already shown that angles H and M are the same.
The sets of corresponding angles are:
H and M, N and I, and O and J
The answer would be angle O and angle J are similar.
What is the length of the diagonal of a 10 cm by 15 cm rectangle
Answer:
5 root 13
Step-by-step explanation:
using the pytogorean therom we get
root(10 squared +15 squared)
=root(100+225)
=root(325)
=5 root 13
Simplify (6ab3c)(-abc2).
6b 2c
-6a 2b 4c 3
-6ab 3c 2
Answer: The simplified version of (6ab3c)(-abc2) is -12a^2 b^4 c^2
Step-by-step explanation:
Answer:
−6a^2b^4c^3
Step-by-step explanation:
how to factorise
25x²- 64y²
Answer:
(5x + 8y)(5x - 8y)
Step-by-step explanation:
Since both terms are perfect squares, factor using the difference of squares formula, a^2 - b^2 = (a + b)(a - b) where a = 5x and b = 8y.
~Hoped this helped~
In the rhombus,
mZ1 =15x, mZ2 = x+y, and m/3 =9z.
Find the value of each variable. The diagram
is not to scale.
2 | 1
a. x = 12, y = 84, z = 20
b. x = 6, y = 174, z = 20
c. X= 6, y = 84, z = 10
d. x = 12, y = 174, z = 10
For the M1 angle, we can conclude the size of angle is 90°. Do the angle equation like this :
15x=90
x = 90/15
x = 6
Do the same thing in M2 :
Subtitute the x value to equation, then evaluate the value by using 90 because it's the same.
x+y=90
6+y=90
y=90-6
y=84
M3 value is same, 90°
9z=90
9z/9 = 90/9
z=10
Hope this helps
Help me with the area of the following triangle??? 25 POINTS question
-13/2x+1 is the answer which the area
How to find it simple:
Do 1/2*3x-4
then use that answer and multiple it with the height which is 2x+1 and you get the answer.
The graph of a linear function is often a parabola.
True or False
A linear function has a graph as a straight line so the given statement is False.
What is a linear function?A straight line on the coordinate plane is represented by a linear function.
A linear function always has the same and constant slope.
The formula for a linear function is f(x) = ax + b, where a and b are real values.
A linear function is given by
y = mx + c where m is the slope and constant.
Since it is constant it means it should be a straight line in coordinate geometry.
Therefore a linear function always makes a straight line, not a parabola.
Hence "A linear function has a graph as a straight line so the given statement is False".
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The face of a clock has a circumference of 14π inches. What is its area?
Answer:
A = 49π
Step-by-step explanation:
C = πd
if d = 14 then r = 7
A = πr²
A = π · 7²
A = 49π
3x - 5 = -3 is the answer positive or negative
Answer:
positive
Step-by-step explanation:
if you time the three by a postive then times that by a negetive you get a negitive if you time by two negitives you get postive
In the expression 3x - 5 = - 3 the value of x is positive 2/3.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, An expression 3x - 5 = - 3.
3x = - 3 + 5.
3x = 2.
x = 2/3 so x is positive.
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which number is composite? 11,2,9,5
Answer:
Answer is 9.
Step-by-step explanation:
Composite number is 9.
I hope it's helpful!
Distribute Terms
-5(x +2)
-5x-10
hope that helps^^
Please help!!
What is the value of Avogado's number?
Answer:
It will be C because I have it in my notes lol
Step-by-step explanation:
Hope this Helped
O is the center of a circle if angle AOB =55° what is angle ACB=?
Answer:
answer is 55/2 which is 27.5 degree
Step-by-step explanation:
angle subtended at center is double the angle subtended on aby other part of the circle...
Raj is thinking of a fraction that is equivalent to 5/8 in which the numerator is greater than 20 and the denominator is less than 50. What is the fraction Raj is thinking about?
Answer:
25/40
Step-by-step explanation:
5 into 8 is 40 and 5 into 5 is 25
Answer:
20 / 32
but the numerator is equal to 20, not greater than 20.
Step-by-step explanation:
5/8 = 10/16 = 20 / 32
Translate X(2,3) Y(0,0) Z(4,4) 2 units right and 3 units up
Answer:
X(4, 6) Y(2, 3) Z(6, 7)
Step-by-step explanation:
If you want to translate something right on a number line, then you are essentially adding to it. Thus you would just add 2 to each of the x values of the coordinates. The same applies with translating something up, all that really means is that you are adding to each of the y values so all you have to do is add 3 to each of the y values and you get your answer.
Answer:
X:(4,6), Y:(2,3), Z:(6,7)
Step-by-step explanation:
2 units right implies a change on the x-axis, meaning you add 2 units to the x of the coordinate pair (x, y). 3 units up implies a change in Y, you apply the same rule and add 3 to the Y of the pair.
2 units right (add 2 to x pair coordinate)
X: (2,3) --> (4,3)
Y: (0,0) --> (2,0)
Z:(4,4) --> (6,4)
and 3 units up (add 3 to y pair coordinate)
X: (4,3) --> (4,6)
Y: (2,0) --> (2,3)
Z: (6,4) --> (6,7)