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zimovet [89]
3 years ago
11

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f(x, y) = 2x2y,

2x2 + 4y2 = 12
Mathematics
1 answer:
hjlf3 years ago
7 0

Answer:

f(-2,-1) = -8 is the minimum value of f(x,y)

f(2,1) = 8 is the maximum value of f(x,y)

Step-by-step explanation:

f(x,y) = 2x²y under the constraint 2x² + 4y² = 12

Let g(x,y) = 2x² + 4y² - 12

df/dx = 4xy, df/dy = 2x², dg/dx = 4x and dg/dy = 8y

Now, using the principle of Lagrange multipliers,

df/dx + λdg/dx = 0 and df/dy + λdg/dy = 0.

Substituting the values of the variables, we have

df/dx + λdg/dx = 0    and df/dy + λdg/dy = 0.

4xy + 4xλ = 0      (1)           2x² + 8λy = 0  (2)

xy + xλ = 0

x(y + λ) = 0

x = 0 or y + λ = 0

Since x ≠ 0, y = -λ

Substituting y = -λ into (2), we have

2x² + 8λy = 0

2x² + 8λ(-λ) = 0

2x² - 8λ² = 0

2x² = 8λ²

x² = 4λ²

x = ±2λ  

Substituting the values of x and y into the constraint equation, we have

2x² + 4y² = 12

2(±2λ)² + 4(-λ)² = 12

2(4)λ² + 4λ² = 12

8λ² + 4λ² = 12

12λ² = 12

λ² = 1

λ = ±1

Substituting the value of  λ into x and y, we have

x = ±2λ = ±2(±1) = ±2

y = -λ = -(±1) = ±1

The minimum values of x and y are -2 and -1 respectively. Substituting these int f(x,y), we have

f(-2,-1) = 2(-2)²(-1) = 2 × 4 × (-1) = -8

So f(-2,-1) = -8 is the minimum value of f(x,y)

The maximum values of x and y are 2 and 1 respectively. Substituting these int f(x,y), we have

f(2,1) = 2(2)²(1) = 2 × 4 × 1 = 8

So f(2,1) = 8 is the maximum value of f(x,y)

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Answer:

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Step-by-step explanation:

Subtract 7 from the first one:

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Now, you can write these as a compound inequality:

  -8 ≤ y ≤ 8

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<em>Additional comment</em>

You basically solve these the same way you would an equation. The only difference is that multiplying or dividing by a negative number will reverse the inequality symbol:

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You have a globe that has a radius of 11 inches. Find the volume of your globe
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Step-by-step explanation:

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2 years ago
I’m confused and don’t have an angle tool to even try to do the question
yanalaym [24]
<h3>Answer:   40</h3>

=================================================

Explanation:

JQ is longer than QN. We can see this visually, but the rule for something like this is the segment from the vertex to the centroid is longer compared to the segment that spans from the centroid to the midpoint.

See the diagram below.

The ratio of these two lengths is 2:1, meaning that JQ is twice as long compared to QN. This is one property of the segments that form when we construct the centroid (recall that the centroid is the intersection of the medians)

We know that JN = 60

Let x = JQ and y = QN

The ratio of x to y is x/y and this is 2/1

x/y = 2/1

1*x = y*2

x = 2y

Now use the segment addition postulate

JQ + QN = JN

x + y = 60

2y + y = 60

3y = 60

y = 60/3

y = 20

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JQ = 2*y = 2*QN = 2*20 = 40

--------------

We have

JQ = 40 and QN = 20

We see that JQ is twice as larger as QN and that JQ + QN is equal to 60.

7 0
3 years ago
What is the value of x?
mariarad [96]
It would be 11 because you would add the x's and divide it by 55. 55/5=11
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