Final answer:
Using Lagrange multipliers, we find the maximum value of the function f(x,y) = x²+y² is 10, occurring at the boundary of the constraint circle, and the minimum is 0, occurring at the center of the circle.
Step-by-step explanation:
We can use Lagrange multipliers to find the maximum and minimum of the function f(x,y) = x²+y² subject to the constraint g(x,y) = x²+y² ≤ 10.
The Lagrange function is L(x, y, λ) = f(x,y) - λ (g(x,y) - 10). Performing partial derivatives with respect to x, y, and λ and setting them equal to zero gives the system of equations:
- ∂L/∂x = 2x - λ(2x) = 0
- ∂L/∂y = 2y - λ(2y) = 0
- ∂L/∂λ = x² + y² - 10 = 0
Solving this system, we find that the Lagrange multiplier, λ, constrains x and y to be on the circle defined by the constraint x² + y² = 10. Since the constraint is a circle with radius √10 and f(x,y) measures distance squared from the origin to a point (x, y), the maximum value of f(x,y) occurs at the boundary of the circle, hence maximum = 10.
The minimum value of f(x,y) is at the center of the circle where x = 0 and y = 0, giving minimum = 0.