Answer :
Use the method of Lagrange multipliers. The minimum value of f(x, y) subject to the constraint x^2 + y^2 = 9 is 747.
To find the minimum value of the function f(x, y) = 83x^2 + 23y^2 subject to the constraint x^2 + y^2 = 9, we can use the method of Lagrange multipliers.
First, we define the function g(x, y) = x^2 + y^2 - 9 as the constraint equation.
Next, we set up the system of equations:
∇f(x, y) = λ∇g(x, y)
(∂f/∂x)i + (∂f/∂y)j = λ((∂g/∂x)i + (∂g/∂y)j)
This gives us the following equations:
166x = λ(2x)
46y = λ(2y)
x^2 + y^2 = 9
From the first two equations, we can solve for x and y in terms of λ:
x = 0 or λ = 83/2
y = 0 or λ = 23/2
Since x^2 + y^2 = 9, we can substitute these values back into the constraint equation to find the corresponding values of λ:
λ = 83/2 → x^2 = 9, y^2 = 0 → x = ±3, y = 0
λ = 23/2 → x^2 = 0, y^2 = 9 → x = 0, y = ±3
Now we can substitute these values of x and y back into the function f(x, y) to find the corresponding values of f:
f(3, 0) = 83(3)^2 + 23(0)^2 = 747
f(-3, 0) = 83(-3)^2 + 23(0)^2 = 747
f(0, 3) = 83(0)^2 + 23(3)^2 = 2079
f(0, -3) = 83(0)^2 + 23(-3)^2 = 2079
Therefore, the minimum value of f(x, y) subject to the constraint x^2 + y^2 = 9 is 747.
Learn more about minimum value from
https://brainly.com/question/30236354
#SPJ11
