Question

Find the indicated maximum or minimum value of f subject to the given constraint. Minimum: f(x, y)=8x² + y²+2xy + 15x +2y: y^² = x+1

Answer 1

To find the indicated minimum value of f subject to the given constraint, use the method of Lagrange multipliers. Set up the equations, find the critical points, and evaluate f(x, y) at these points to determine the minimum value.

Step-by-step explanation:

To find the indicated minimum value of f subject to the given constraint, we can use the method of Lagrange multipliers. First, we need to find the critical points of the function f(x, y) = 8x² + y² + 2xy + 15x + 2y. To do this, we need to solve the system of equations formed by setting the partial derivatives of f equal to zero and the constraint equation. After finding the critical points, we evaluate f(x, y) at these points to determine the minimum value.

Here are the steps:

1. Set up the equations: ∇f = λ∇g and g = 0 (where ∇f is the gradient of f, ∇g is the gradient of the constraint equation y² = x + 1, and λ is the Lagrange multiplier).
2. Find the partial derivatives of f(x, y): ∂f/∂x = 16x + 2y + 15, ∂f/∂y = 2x + 2 + 2y.
3. Find the partial derivatives of the constraint equation: ∂g/∂x = -1, ∂g/∂y = 2y.
4. Set up the equations using the partial derivatives: 16x + 2y + 15 = λ(-1), 2x + 2 + 2y = λ(2y).
5. Solve the system of equations for x, y, and λ. You can do this by substituting one equation into the other and solving for the variables.
6. Once you find the critical points, evaluate f(x, y) at these points to find the minimum value.