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
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)

You might be interested in
The given series has six terms. what is the sum of the terms of the series? 17 + 25 + 33 + . . . + 57
nikdorinn [45]
Sum of the terms of the series is

Sn = n/2 ( a1+an )

we have n= 6 , a1= 17, an = 57
so Sn = 6/2 ( 17+57) = 3(74) = 222
3 0
2 years ago
List all the 4 digit numbers that fit these clues
natita [175]
If you give the clues I could help.
4 0
3 years ago
Simplify.<br><br> 8−4÷2+5⋅6<br><br> 1. 32<br> 2. 36<br> 3. 54<br> 4. 66
Radda [10]

Answer: 36

Step-by-step explanation:

8-2+30

6+30

36

7 0
2 years ago
M is the midpoint of AB if A (4,0) and M  ( 2, 1.5) then find the coordinates of B
REY [17]

Answer:

C. (0, 3)

Step-by-step explanation:

3 0
2 years ago
What equation is equivalent to 1/3 (9x+12) + 4x + 6
Irina-Kira [14]
The answer is b, 7x+10
3 0
2 years ago
Read 2 more answers
Other questions:
  • Garrick gets paid $4.70 an hour with time-and-a-half for overtime (over 40 hours) How much did he earn one week when he worked 4
    5·1 answer
  • Which statement is not true of dot plots?
    10·2 answers
  • Which equations have the same value of x as 2/3(6x+12)=-24 Select two options.
    5·2 answers
  • Abigail needs 19th inches of copper for an experiment. the wire is sold by the centimeter given that 1 inch equals 2.54 cm how m
    13·1 answer
  • A. -6<br> B. -1<br> C. 1<br> D. 6<br><br> Please help quizzz
    14·2 answers
  • The domain of the function is
    6·1 answer
  • On a math test, 5 out of 20 students got an A. If three students are chosen at random without replacement, what is the probabili
    13·1 answer
  • PLEASE HELP ME! PLEASEEE
    5·2 answers
  • Find the missing side lengths leave your answer as a racials simplest form
    9·1 answer
  • What is the area of the regular hexagon
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!