(GIVING BRAINLIEST!!!)

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer d

andrey2020 [161]

Answer:

The maximum value is 1/27 and the minimum value is 0.

Step-by-step explanation:

Note that the given function is equal to $(xyz)^2$ then it means that it is positive i.e $f(x,y,z)\geq 0$ .

Consider the function $F(x,y,z,\lambda)=x^2y^2z^2-\lambda (x^2+y^2+z^2-1)$

We want that the gradient of this function is equal to zero. That is (the calculations in between are omitted)

$\frac{\partial F}{\partial x} = 2x(y^2z^2 - \lambda)=0$

$\frac{\partial F}{\partial y} = 2y(x^2z^2 - \lambda)=0$

$\frac{\partial F}{\partial z} = 2z(x^2y^2 - \lambda)=0$

$\frac{\partial F}{\partial \lambda} = (x^2+y^2+z^2-1)=0$

Note that the last equation is our restriction. The restriction guarantees us that at least one of the variables is non-zero. We've got 3 options, either 1, 2 or none of them are zero.

If any of them is zero, we have that the value of the original function is 0. We just need to check that there exists a value for lambda.

Suppose that x is zero. Then, from the second and third equation we have that

$-2y\lambda = -2z\lambda$ . If lambda is not zero, then y =z. But, since $-2y\lambda=0$ and lambda is not zero, this implies that x=y=z=0 which is not possible. This proofs that if one of the variables is 0, then lambda is zero. So, having one or two variables equal to zero are feasible solutions for the problem.

Suppose that only x is zero, then we have the solution set $y^2+z^2=1$ .

If both x,y are zero, then we have the solution set $z^2=1$ . We can find the different solution sets by choosing the variables that are set to zero.

NOw, suppose that none of the variables are zero.

From the first and second equation we have that

$\lambda = y^2z^2 = x^2z^2$ which implies $x^2=y^2$

Also, from the first and third equation we have that

$\lambda = y^2z^2 = x^2y^2$ which implies $x^2=z^2$

So, in this case, replacing this in the restriction we have $3z^2=1$ , which gives as another solution set. On this set, we have $x^2=y^2=z^2=\frac{1}{3}$ . Over this solution set, we have that the value of our function is $\frac{1}{3^3}= \frac{1}{27}$

4 0

3 years ago

Matt gets paid an hourly rate of $12.50 for the first 40 hours he works in a week. After 40 hours , Matt gets paid at an overtim

lukranit [14]

12.50 x 40 = 500; this is his first week pay from $12.50 an hour for 40 hours

(15 x 55) + 100 = 925; this is the second pay from $15 an hour for 55 hours and an additional $100

Hope this helps! (If not correct me if its wrong xd)

5 0

3 years ago

(CHALLENGE QUESTION FOR ALL OF YOU):

NARA [144]

Answer:

Euler's Identity

Euler's Identity is written simply as: e^(iπ) + 1 = 0, it comprises the five most important mathematical constants, and it is an equation that has been compared to a Shakespearean sonnet.

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers

SOMEONE HELP ON NUMBER 1

vredina [299]

Answer:

It is most definitely D, The volume of a cone is one-third of the volume of a cylinder.

8 0

3 years ago

Help 2,4+0,8=.......-0,06=........+1,21=.......+1,78=....-5,14=.

horrorfan [7]

Lets write this out:-

2.4 + 0.8 =______ - 0.06 = _____+ 1.21=_____+1.78=_____- 5.14=___

So to solve d blanks we will do d following:-

2.4 + 0.8 = 3.2

Now lets write this out AGAIN.

2.4 + 0.8 = 3.2 - 0.06 = _____+ 1.21=_____+1.78=_____- 5.14=___

Now lets solve again:-

3.2 - 0.06 = 3.14

Now lets write this out AGAIN.

2.4 + 0.8 = 3.2 - 0.06 = 3.14 + 1.21=_____+1.78=_____- 5.14=___

Now lets solve again:-

3.14 + 1.21= 4.25

Now lets write this out AGAIN.

2.4 + 0.8 = 3.2 - 0.06 = 3.14 + 1.21 = 4.25 +1.78=_____- 5.14=___

Now lets solve again:-

4.25 +1.78= 6.03

Now lets write this out AGAIN.

2.4 + 0.8 = 3.2 - 0.06 = 3.14 + 1.21 = 4.25 +1.78= 6.03 - 5.14=___

Now lets solve again:-

6.03 - 5.14 = 0.89

Now lets write this out AGAIN.

2.4 + 0.8 = 3.2 - 0.06 = 3.14 + 1.21 = 4.25 +1.78= 6.03 - 5.14= 0.89

So, 2.4 + 0.8 = 3.2 - 0.06 = 3.14 + 1.21 = 4.25 +1.78= 6.03 - 5.14= 0.89

Hope I helped ya!! xD

3 0

3 years ago