Answer :
The maximum value of f(x, y) = 83 - x^2 - y^2 subject to the constraint x + y - 2 = 0 is f(2, 2) = 75. To find the maximum value of the function f(x, y) = 83 - x^2 - y^2 subject to the constraint x + y - 2 = 0, we can use the method of Lagrange multipliers.
The first step is to set up the Lagrangian function L(x, y, λ) as follows:
L(x, y, λ) = f(x, y) - λ(g(x, y))
where g(x, y) is the constraint equation, and λ is the Lagrange multiplier.
In this case, the constraint equation is x + y - 2 = 0. So, the Lagrangian function becomes:
L(x, y, λ) = (83 - x^2 - y^2) - λ(x + y - 2)
Next, we need to find the critical points of L(x, y, λ) by taking the partial derivatives with respect to x, y, and λ, and setting them equal to zero:
∂L/∂x = -2x - λ = 0
∂L/∂y = -2y - λ = 0
∂L/∂λ = x + y - 2 = 0
From the first two equations, we can solve for x and y in terms of λ:
-2x - λ = 0 => x = -λ/2
-2y - λ = 0 => y = -λ/2
Substituting these values into the third equation, we get:
x + y - 2 = 0
=> (-λ/2) + (-λ/2) - 2 = 0
=> -λ - 4 = 0
=> λ = -4
Substituting this value of λ back into the expressions for x and y, we get:
x = -(-4)/2 = 2
y = -(-4)/2 = 2
Therefore, the critical point is (2, 2).
To determine if this critical point is a maximum or minimum, we need to examine the second partial derivatives of L(x, y, λ).
The second partial derivatives are:
∂²L/∂x² = -2
∂²L/∂y² = -2
∂²L/∂x∂y = 0
Calculating the determinant of the Hessian matrix (the matrix of second partial derivatives):
D = (∂²L/∂x²)(∂²L/∂y²) - (∂²L/∂x∂y)²
=> D = (-2)(-2) - (0)²
=> D = 4
Since the determinant is positive, and the second partial derivative with respect to x is negative, the critical point (2, 2) corresponds to a maximum value of the function f(x, y).
Therefore, the maximum value of f(x, y) = 83 - x^2 - y^2 subject to the constraint x + y - 2 = 0 is f(2, 2) = 83 - 2^2 - 2^2 = 83 - 4 - 4 = 75.
Learn more about Lagrangian function from the given link:
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