Answer:
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 = [tex]= \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}\\[/tex]
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
