Final answer:
To use Lagrange multipliers to find the maximum and minimum values of the function given the constraint, set up a system of equations by equating the gradients of the function and the constraint and introduce a Lagrange multiplier. Solve for the variables, evaluate the function at these points, and consider boundary points. Use the Second Derivative Test for confirmation of the extremities.
Step-by-step explanation:
To find the maximum and minimum values of the function f(x,y)=(x+6)²+(y−6)² subject to the constraint x²+y²≤50 using Lagrange multipliers, we introduce a Lagrange multiplier λ and set up the following system of equations by taking the gradient of the function and the constraint:
- ∂f/∂x = 2(x+6) = λ*2x
- ∂f/∂y = 2(y-6) = λ*2y
- x² + y² = 50
From these equations, we solve for x and y in terms of the Lagrange multiplier λ. We can find the values of λ that correspond to the intersection points of the function f(x,y) and the constraint circle. After solving for x and y, we plug these values into f(x,y) to find the maximum and minimum values. However, the critical points may also lie on the boundary of the constraint, in which case we would also evaluate the function along the circle x² + y² = 50.
To determine if the values obtained are indeed a maximum or minimum, we can use the Second Derivative Test by computing the Hessian Matrix at those critical points and analyze its definiteness. The maximum and minimum values (when they exist) are the largest and smallest values obtained from evaluating the function at these critical points and boundary points.