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)
