Final answer:
To solve the optimization problem using Lagrange multipliers, set up the Lagrange function by defining f(x, y), g(x, y), and λ. Differentiate the Lagrange function with respect to x, y, and λ, and set the derivatives equal to 0.
Step-by-step explanation:
To solve this optimization problem using Lagrange multipliers, we first need to set up the Lagrange function. The Lagrange function is defined as L(x, y, λ) = f(x, y) - λ(g(x, y) - c), where f(x, y) is the function to be minimized, g(x, y) is the equality constraint, and λ is the Lagrange multiplier. In this case, f(x, y) = x^2 + y^2 and g(x, y) = x + 2y - 35. So, the Lagrange function becomes L(x, y, λ) = x^2 + y^2 - λ(x + 2y - 35).
Next, we differentiate the Lagrange function with respect to x, y, and λ, and set the derivatives equal to 0 to find critical points. Taking the derivatives, we get ∂L/∂x = 2x - λ = 0, ∂L/∂y = 2y - 2λ = 0, and ∂L/∂λ = x + 2y - 35 = 0.
Solving these three equations simultaneously, we can find the values of x, y, and λ. Plugging these values back into the original function f(x, y) = x^2 + y^2, we can find the minimum value of f(x, y) that satisfies the given constraint.