Mathematics, 27.08.2020 01:01 addisynshepherd

The function f(x, y) = xy has an absolute maximum value and absolute minimum value subject to the constraint 2x^2 + 3y^2 - 3xy = 49. Use Lagrange multipliers to find these values. A. Find the gradient of f(x, y) = 3xy.
B. Find the gradient of g(x, y) = x2 + y2 - xy-9.
C. Write the Lagrange multiplier conditions.
A. 3xy = M2x - y), 3xy = M2y - x), x2 + y2 - xy - 9 = 0
B. 3x = M(2x - y), 3y = M2y - x), 3xy = 0
C. 3y = M(2x - y), 3x = M(2y – x), x² + y2 - xy - 9 = 0
D. 3x = M(2x - y), 3y = M2y - x), x2 + y2 - xy - 9 = 0
The absolute maximum value is:.
The absolute minimum value is:.

Answers: 2

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Answer from: blondesaturn

The absolute minimum of f(x,y) = 8.107

The absolute maximum of f(x,y) = 24.326

Step-by-step explanation:

f(x,y) = xy. The constraint equation is 2x² + 3y² - 3xy = 49

Let g(x,y) = 2x² + 3y² - 3xy - 49

df/dx = y and df/dy = x , dg/dx = 4x - 3y and dg/dy = 6y - 3x

Using Lagrange multipliers,

df/dx = λdg/dx and df/dy = λdg/dy

So,

y = λ(4x - 3y) (1 )and x = λ(6y - 3x) (2)

y = 4λx - 3λy

y + 3λy = 4λx

y(1 + 3λ) = 4λx

y = 4λx/(1 + 3λ)

Substituting y into (2), we have

x = λ(6y - 3x)

x = λ(6[4λx/(1 + 3λ)] - 3x)

x = 24λ²x/(1 + 3λ) - 3λx

24λ²x/(1 + 3λ) - 3λx - x = 0

[24λ²/(1 + 3λ) - 3λ - 1]x = 0

⇒ [24λ²/(1 + 3λ) - 3λ - 1] = 0 since x ≠ 0

[24λ²/(1 + 3λ) - 3λ - 1] = 0

⇒[24λ²/(1 + 3λ) - (3λ + 1)] = 0

[24λ² - (3λ + 1)²] = 0

24λ² - 9λ² - 6λ - 1 = 0

15λ² - 6λ - 1 = 0

Using the quadratic formula,

λ = $= \frac{-(-6) +/- \sqrt{(-6)^{2} - 4 X 15 X (-1)} }{2 X 15}\\= \frac{6) +/- \sqrt{36 + 60)} }{30}\\= \frac{6 +/- \sqrt{96)} }{30}\\= \frac{6 +/- 4\sqrt{6)} }{30}\\$

λ = (6 + 4√6)/30 or (6 - 4√6)/30

λ = (3 + 2√6)/15 = 0.527 or (3 - 2√6)/15 = -0.127

Substituting y into the constraint equation, we have

2x² + 3y² - 3xy = 49

2x² + 3(4λx/(1 + 3λ))² - 3x(4λx/(1 + 3λ)) = 49

2x² + 12λ²x²/(1 + 3λ))² - 12λx²/(1 + 3λ) = 49

[2 + 12λ²/(1 + 3λ)² - 12λ/(1 + 3λ)}x² = 49

[2(1 + 3λ)² + 12λ² - 12λ(1 + 3λ)]x²/(1 + 3λ)² = 49

[2(1 + 6λ + 9λ²) + 12λ² - 12λ + 36λ²)]x²/(1 + 3λ)² = 49

[2 + 12λ + 18λ² + 12λ² - 12λ + 36λ²)]x²/(1 + 3λ)² = 49

[2 + 6λ²]x²/(1 + 3λ)² = 49

x² = 49(1 + 3λ)²/(2 + 6λ²)

x² = 49(1 + 3λ)²/2(1 + 3λ²)

x = √[49(1 + 3λ)²/2(1 + 3λ²)]

x = ±7√[(1 + 3λ)²/2(1 + 3λ²)]

Substituting λ = (3 + 2√6)/15 = 0.527 or (3 - 2√6)/15 = -0.127

x = ±7√[(1 + 3(0.527))²/2(1 + 3(0.527)²)] or ±7√[(1 + 3(-0.127))²/2(1 + 3(-0.127)²)]

x = ±7√[(6.662/3.666] or ±7√[0.3831/1.9032)]

x = ±7√1.8172 or ±7√0.2012

x = ±9.436 or ±3.141

Substituting x and λ into y, we have

y = 4λx/(1 + 3λ)

y = 4(0.527)(±3.141)/(1 + 3(0.527)) or 4(-0.127)(±3.141)/(1 + 3(-0.127))

y = ±6.6621/2.581 or ±1.5956/0.619

y = ±2.581 or ±2.578

The minimum value of f(x,y) is gotten at the minimum values of x and y which are x = -3.141 and y = -2.581

So f(-3.141,-2.581) = -3.141 × -2.581 = 8.107

The maximum value of f(x,y) is gotten at the minimum values of x and y which are x = +9.436 and y = +2.578

So f(+9.436,+2.578) = +9.436 × +2.578 = 24.326

Answer from: Quest

answer: it's d

step-by-step explanation:

Answer from: Quest

answer: yard ramps with a level off platform are typically 36 feet long. truck, rather than at an angle, enabling access to those pallets at the very back of the truck. if you are using your yard ramp to convert a loading dock into a drive-in door, we even at 60 inches (5 feet) high and a 30 foot ramp, the slope of the ramp will be less

step-by-step explanation:

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The function f(x, y) = xy has an absolute maximum value and absolute minimum value subject to the co...